Submodules and Cosets of Submodules in Right Module over Associative Ring

by
Michal Muzalewski, and
Wojciech Skaba

Copyright (c) 1990 Association of Mizar Users

MML identifier: RMOD_2
[ MML identifier index ]

environ

vocabulary FUNCSDOM, VECTSP_1, VECTSP_2, RLSUB_1, BOOLE, RLVECT_1, ARYTM_1,
LMOD_4, RELAT_1, FUNCT_1, BINOP_1;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, RELAT_1, FUNCT_1, FUNCT_2,
STRUCT_0, DOMAIN_1, RLVECT_1, BINOP_1, VECTSP_1, FUNCSDOM, VECTSP_2;
constructors DOMAIN_1, BINOP_1, VECTSP_2, PARTFUN1, MEMBERED, XBOOLE_0;
clusters FUNCT_1, VECTSP_2, STRUCT_0, RELSET_1, SUBSET_1, MEMBERED, ZFMISC_1,
XBOOLE_0;
requirements SUBSET, BOOLE;
definitions TARSKI, VECTSP_2, RLVECT_1, XBOOLE_0;
theorems BINOP_1, FUNCT_1, FUNCT_2, TARSKI, VECTSP_1, ZFMISC_1, VECTSP_2,
MOD_1, RLVECT_1, RELAT_1, RELSET_1, XBOOLE_0, XBOOLE_1;
schemes XBOOLE_0;

begin

reserve x,y,y1,y2 for set;
reserve R for Ring;
reserve a for Scalar of R;
reserve V,X,Y for RightMod of R;
reserve u,u1,u2,v,v1,v2 for Vector of V;
reserve V1,V2,V3 for Subset of V;

definition let R, V, V1;
attr V1 is lineary-closed means :Def1:
(for v,u st v in V1 & u in V1 holds v + u in V1) &
(for a,v st v in V1 holds v * a in V1);
end;

canceled 3;

theorem Th4:
V1 <> {} & V1 is lineary-closed implies 0.V in V1
proof assume that A1: V1 <> {} and A2: V1 is lineary-closed;
consider x being Element of V1;
reconsider x as Element of V by A1,TARSKI:def 3;
x * 0.R in V1 by A1,A2,Def1;
hence thesis by MOD_1:37;
end;

theorem Th5:
V1 is lineary-closed implies (for v st v in V1 holds - v in V1)
proof assume A1: V1 is lineary-closed;
let v; assume v in V1;
then v * (- 1_ R) in V1 by A1,Def1;
hence thesis by MOD_1:37;
end;

theorem
V1 is lineary-closed implies
(for v,u st v in V1 & u in V1 holds v - u in V1)
proof assume A1: V1 is lineary-closed;
let v,u; assume that A2: v in V1 and A3: u in V1;
v - u = v + (- u) & - u in V1 by A1,A3,Th5,RLVECT_1:def 11;
hence thesis by A1,A2,Def1;
end;

theorem Th7:
{0.V} is lineary-closed
proof
thus for v,u st v in {0.V} & u in {0.V} holds v + u in {0.V}
proof let v,u;
assume v in {0.V} & u in {0.V};
then v = 0.V & u = 0.V by TARSKI:def 1;
then v + u = 0.V & 0.V in {0.V} by TARSKI:def 1,VECTSP_1:7;
hence thesis;
end;
let a,v;
assume A1: v in {0.V};
then v = 0.V by TARSKI:def 1;
hence thesis by A1,MOD_1:37;
end;

theorem
the carrier of V = V1 implies V1 is lineary-closed
proof assume A1: the carrier of V = V1;
hence for v,u st v in V1 & u in V1 holds v + u in V1;
let a,v;
assume v in V1;
thus v * a in V1 by A1;
end;

theorem
V1 is lineary-closed & V2 is lineary-closed &
V3 = {v + u : v in V1 & u in V2} implies V3 is lineary-closed
proof assume that A1: V1 is lineary-closed & V2 is lineary-closed and
A2: V3 = {v + u : v in V1 & u in V2};
thus for v,u st v in V3 & u in V3 holds v + u in V3
proof let v,u;
assume that A3: v in V3 and A4: u in V3;
consider v1,v2 such that A5: v = v1 + v2 and
A6: v1 in V1 & v2 in V2 by A2,A3;
consider u1,u2 such that A7: u = u1 + u2 and
A8: u1 in V1 & u2 in V2 by A2,A4;
A9: v1 + u1 in V1 & v2 + u2 in V2 by A1,A6,A8,Def1;
v + u = ((v1 + v2) + u1) + u2 by A5,A7,RLVECT_1:def 6
.= ((v1 + u1) + v2) + u2 by RLVECT_1:def 6
.= (v1 + u1) + (v2 + u2) by RLVECT_1:def 6;
hence thesis by A2,A9;
end;
let a,v;
assume v in V3;
then consider v1,v2 such that A10: v = v1 + v2 and
A11: v1 in V1 & v2 in V2 by A2;
A12: v1 * a in V1 & v2 * a in V2 by A1,A11,Def1;
v * a = v1 * a + v2 * a by A10,VECTSP_2:def 23;
hence v * a in V3 by A2,A12;
end;

theorem
V1 is lineary-closed & V2 is lineary-closed implies
V1 /\ V2 is lineary-closed
proof assume A1: V1 is lineary-closed & V2 is lineary-closed;
thus for v,u st v in V1 /\ V2 & u in V1 /\ V2 holds v + u in V1 /\ V2
proof let v,u;
assume v in V1 /\ V2 & u in V1 /\ V2;
then v in V1 & v in V2 & u in V1 & u in V2 by XBOOLE_0:def 3;
then v + u in V1 & v + u in V2 by A1,Def1;
hence thesis by XBOOLE_0:def 3;
end;
let a,v;
assume v in V1 /\ V2;
then v in V1 & v in V2 by XBOOLE_0:def 3;
then v * a in V1 & v * a in V2 by A1,Def1;
hence thesis by XBOOLE_0:def 3;
end;

definition let R; let V;
mode Submodule of V -> RightMod of R means :Def2:
the carrier of it c= the carrier of V &
the Zero of it = the Zero of V &
[:the carrier of it,the carrier of it:] &
the rmult of it =
(the rmult of V) | [:the carrier of it, the carrier of R:];
existence
proof take V;
thus the carrier of V c= the carrier of V &
the Zero of V = the Zero of V;
thus thesis by FUNCT_2:40;
end;
end;

reserve W,W1,W2 for Submodule of V;
reserve w,w1,w2 for Vector of W;

canceled 5;

theorem
x in W1 & W1 is Submodule of W2 implies x in W2
proof assume x in W1 & W1 is Submodule of W2;
then x in the carrier of W1 &
the carrier of W1 c= the carrier of W2 by Def2,RLVECT_1:def 1;
hence thesis by RLVECT_1:def 1;
end;

theorem Th17:
x in W implies x in V
proof assume x in W;
then x in the carrier of W &
the carrier of W c= the carrier of V by Def2,RLVECT_1:def 1;
hence thesis by RLVECT_1:def 1;
end;

theorem Th18:
w is Vector of V
proof w in W by RLVECT_1:3;
then w in V by Th17;
hence thesis by RLVECT_1:def 1;
end;

theorem Th19:
0.W = 0.V
proof
thus 0.W = the Zero of W by RLVECT_1:def 2
.= the Zero of V by Def2
.= 0.V by RLVECT_1:def 2;
end;

theorem
0.W1 = 0.W2
proof
thus 0.W1 = 0.V by Th19
.= 0.W2 by Th19;
end;

theorem Th21:
w1 = v & w2 = u implies w1 + w2 = v + u
proof assume A1: v = w1 & u = w2;
set IW = [:the carrier of W, the carrier of W:];
reconsider ww1 = w1, ww2 = w2 as Vector of V by Th18;
A2: v + u = (the add of V).(ww1,ww2) by A1,RLVECT_1:5
.= (the add of V).[ww1,ww2] by BINOP_1:def 1;
w1 + w2 = (the add of W).(w1,w2) by RLVECT_1:5
.= ((the add of V) | IW).(w1,w2) by Def2
.= ((the add of V) | IW).[w1,w2] by BINOP_1:def 1;
hence thesis by A2,FUNCT_1:72;
end;

theorem Th22:
w = v implies w * a = v * a
proof assume A1: w = v;
reconsider ww1 = w as Vector of V by Th18;
A2: v * a = (the rmult of V).(v,a) by VECTSP_2:def 15
.= (the rmult of V).[ww1,a] by A1,BINOP_1:def 1;
w * a = (the rmult of W).(w,a) by VECTSP_2:def 15
.= (the rmult of W).[w,a] by BINOP_1:def 1
.= ((the rmult of V) | ([:the carrier of W, the carrier of R:])
).[w,a] by Def2;
hence thesis by A2,FUNCT_1:72;
end;

theorem Th23:
w = v implies - v = - w
proof assume A1: w = v;
- v = v * (- 1_ R) & - w = w * (- 1_ R) by MOD_1:37;
hence thesis by A1,Th22;
end;

theorem Th24:
w1 = v & w2 = u implies w1 - w2 = v - u
proof assume that A1: w1 = v and A2: w2 = u;
A3: - w2 = - u by A2,Th23;
w1 - w2 = w1 + (- w2) & v - u = v + (- u) by RLVECT_1:def 11;
hence thesis by A1,A3,Th21;
end;

Lm1: the carrier of W = V1 implies V1 is lineary-closed
proof assume A1: the carrier of W = V1;
set VW = the carrier of W;
reconsider WW = W as RightMod of R;
thus for v,u st v in V1 & u in V1 holds v + u in V1
proof let v,u;
assume v in V1 & u in V1;
then reconsider vv = v, uu = u as Vector of WW by A1;
reconsider vw = vv + uu as Element of VW;
vw in V1 by A1;
hence v + u in V1 by Th21;
end;
let a,v;
assume v in V1;
then reconsider vv = v as Vector of WW by A1;
reconsider vw = vv * a as Element of VW;
vw in V1 by A1;
hence v * a in V1 by Th22;
end;

theorem Th25:
0.V in W
proof 0.W in W & 0.V = 0.W by Th19,RLVECT_1:3;
hence thesis;
end;

theorem
0.W1 in W2
proof 0.W1 = 0.V by Th19;
hence thesis by Th25;
end;

theorem
0.W in V
proof 0.W in W by RLVECT_1:3;
hence thesis by Th17;
end;

theorem Th28:
u in W & v in W implies u + v in W
proof assume u in W & v in W;
then A1: u in the carrier of W &
v in the carrier of W by RLVECT_1:def 1;
reconsider VW = the carrier of W as Subset of V by Def2;
VW is lineary-closed by Lm1;
then u + v in the carrier of W by A1,Def1;
hence thesis by RLVECT_1:def 1;
end;

theorem Th29:
v in W implies v * a in W
proof assume v in W;
then A1: v in the carrier of W by RLVECT_1:def 1;
reconsider VW = the carrier of W as Subset of V by Def2;
VW is lineary-closed by Lm1;
then v * a in the carrier of W by A1,Def1;
hence thesis by RLVECT_1:def 1;
end;

theorem Th30:
v in W implies - v in W
proof assume v in W;
then v * (- 1_ R) in W by Th29;
hence thesis by MOD_1:37;
end;

theorem Th31:
u in W & v in W implies u - v in W
proof assume that A1: u in W and A2: v in W;
- v in W by A2,Th30;
then u + (- v) in W by A1,Th28;
hence thesis by RLVECT_1:def 11;
end;

theorem Th32:
V is Submodule of V
proof
A1: the carrier of V c= the carrier of V &
the Zero of V = the Zero of V;
[:the carrier of V, the carrier of V:] &
the rmult of V =
(the rmult of V) | [:the carrier of V, the carrier of R:]
by FUNCT_2:40;
hence thesis by A1,Def2;
end;

theorem Th33:
for X,V being strict RightMod of R
holds V is Submodule of X & X is Submodule of V implies V = X
proof let X,V be strict RightMod of R;
assume A1: V is Submodule of X & X is Submodule of V;
set VV = the carrier of V;
set VX = the carrier of X;
set AV = the add of V;
set AX = the add of X;
set MV = the rmult of V; set MX = the rmult of X;
VV c= VX & VX c= VV by A1,Def2;
then A2: VV = VX by XBOOLE_0:def 10;
A3: the Zero of V = the Zero of X by A1,Def2;
AV = AX | [:VV,VV:] & AX = AV | [:VX,VX:] by A1,Def2;
then A4: AV = AX by A2,RELAT_1:101;
MV = MX | [:VV,the carrier of R:] &
MX = MV | [:VX,the carrier of R:] by A1,Def2;
hence thesis by A2,A3,A4,RELAT_1:101;
end;

definition let R,V;
cluster strict Submodule of V;
existence
proof
set M = the rmult of V;
set W = RightModStr (# the carrier of V,the add of V, 0.V,M #);
A1: now let a,b be Element of W;
let x,y be Vector of V;
assume A2: x = a & b = y;
thus a + b = (the add of W).(a,b) by RLVECT_1:5
.= x + y by A2,RLVECT_1:5;
end;
A3:  W is Abelian add-associative right_zeroed right_complementable
proof
thus W is Abelian
proof
let a,b be Element of W;
reconsider x = a, y = b as Vector of V;
thus a + b = y + x by A1
.= b + a by A1;
end;
hereby let a,b,c be Element of W;
reconsider x = a, y = b, z = c as Vector of V;
A4: a + b = x + y & b + c = y + z by A1;
hence a + b + c = x + y + z by A1
.= x + (y + z) by RLVECT_1:def 6
.= a + (b + c) by A1,A4;
end;
A5: 0.W = 0.V by RLVECT_1:def 2;
hereby let a be Element of W;
reconsider x = a as Vector of V;
thus a + 0.W = x + 0.V by A1,A5
.= a by VECTSP_1:7;
end;
let a be Element of W;
reconsider x = a as Vector of V;
reconsider b = (comp V).x as Element of W;
take b;
thus a + b = x + (comp V).x by A1
.= x + - x by VECTSP_1:def 25
.= 0.W by A5,RLVECT_1:16;
end;
W is RightMod-like
proof let a,b be Scalar of R;
let v,w be Vector of W;
reconsider x = v, y = w as Vector of V;
A6: now let a be Scalar of R;
let x be Element of W;
let y be Vector of V;
assume y = x;
hence x * a = M.(y,a) by VECTSP_2:def 15
.= y * a by VECTSP_2:def 15;
end;
then A7: v * b = x * b & v * a = x * a & w * a = y * a;
then v * a + w * a = x * a + y * a & v + w = x + y by A1;
hence (v + w) * a = (x + y) * a by A6
.= x * a + y * a by VECTSP_2:def 23
.= v * a + w * a by A1,A7;
thus v * (a + b) = x * (a + b) by A6
.= x * a + x * b by VECTSP_2:def 23
.= v * a + v * b by A1,A7;
thus v * (b * a) = x * (b * a) by A6
.= x * b * a by VECTSP_2:def 23
.= v * b * a by A6,A7;
thus v * 1_ R = x * 1_ R by A6
.= v by VECTSP_2:def 23;
end;
then reconsider W as RightMod of R by A3;
A8:   the Zero of W = the Zero of V by RLVECT_1:def 2;
the add of W = (the add of V) | [:the carrier of W,the carrier of W:] &
the rmult of W =
(the rmult of V) | [:the carrier of W, the carrier of R:]
by FUNCT_2:40;
then reconsider W as Submodule of V by A8,Def2;
take W;
thus thesis;
end;
end;

theorem Th34:
V is Submodule of X & X is Submodule of Y implies V is Submodule of Y
proof assume A1: V is Submodule of X & X is Submodule of Y;
A2: the carrier of V c= the carrier of Y
proof
the carrier of V c= the carrier of X &
the carrier of X c= the carrier of Y by A1,Def2;
hence thesis by XBOOLE_1:1;
end;
A3: the Zero of V = the Zero of Y
proof the Zero of V = the Zero of X &
the Zero of X = the Zero of Y by A1,Def2;
hence thesis;
end;
A4: the add of V =
[:the carrier of V, the carrier of V:]
proof set AV = the add of V;
set VV = the carrier of V;
set AX = the add of X;
set VX = the carrier of X;
set AY = the add of Y;
AV = AX | [:VV,VV:] & AX = AY | [:VX,VX:] & VV c= VX
by A1,Def2;
then AV = (AY | [:VX,VX:]) | [:VV,VV:] & [:VV,VV:] c= [:VX,VX:]
by ZFMISC_1:119;
hence thesis by FUNCT_1:82;
end;
set MV = the rmult of V; set VV = the carrier of V;
set MX = the rmult of X; set VX = the carrier of X;
set MY = the rmult of Y;
MV = MX | [:VV,the carrier of R:] & MX = MY | [:VX,the carrier of R:] &
VV c= VX by A1,Def2;
then MV = (MY | [:VX,the carrier of R:]) | [:VV,the carrier of R:] &
[:VV,the carrier of R:] c=
[:the carrier of X,the carrier of R:] by ZFMISC_1:118;
then MV = MY | [:VV,the carrier of R:] by FUNCT_1:82;
hence thesis by A2,A3,A4,Def2;
end;

theorem Th35:
the carrier of W1 c= the carrier of W2 implies
W1 is Submodule of W2
proof assume
A1: the carrier of W1 c= the carrier of W2;
set VW1 = the carrier of W1;
set VW2 = the carrier of W2;
set MW1 = the rmult of W1; set MW2 = the rmult of W2;
set AV = the add of V; set MV = the rmult of V;
A2: the Zero of W1 = the Zero of V &
the Zero of W2 = the Zero of V by Def2;
A3: the add of W1 =
[:the carrier of W1,
the carrier of W1:]
proof
(the add of W1) = AV | [:VW1,VW1:] & (the add of W2)
= AV | [:VW2,VW2:] &
[:VW1,VW1:] c= [:VW2,VW2:] by A1,Def2,ZFMISC_1:119;
hence thesis by FUNCT_1:82;
end;
MW1 = MV | [:VW1,the carrier of R:] &
MW2 = MV | [:VW2,the carrier of R:] &
[:VW1,the carrier of R:] c= [:VW2,the carrier of R:]
by A1,Def2,ZFMISC_1:118;
then MW1 = MW2 | [:VW1,the carrier of R:] by FUNCT_1:82;
hence thesis by A1,A2,A3,Def2;
end;

theorem
(for v st v in W1 holds v in W2) implies W1 is Submodule of W2
proof assume A1: for v st v in W1 holds v in W2;
the carrier of W1 c= the carrier of W2
proof let x be set;
assume A2: x in the carrier of W1;
the carrier of W1 c= the carrier of V by Def2;
then reconsider v = x as Vector of V by A2;
v in W1 by A2,RLVECT_1:def 1;
then v in W2 by A1;
hence thesis by RLVECT_1:def 1;
end;
hence thesis by Th35;
end;

theorem Th37:
for W1,W2 being strict Submodule of V
holds the carrier of W1 = the carrier of W2 implies
W1 = W2
proof let W1,W2 be strict Submodule of V;
assume the carrier of W1 =
the carrier of W2;
then W1 is Submodule of W2 & W2 is Submodule of W1 by Th35;
hence thesis by Th33;
end;

theorem Th38:
for W1,W2 being strict Submodule of V
holds (for v being Vector of V holds v in W1 iff v in W2) implies W1 = W2
proof let W1,W2 be strict Submodule of V;
assume A1: for v being Vector of V holds v in W1 iff v in W2;
x in the carrier of W1 iff
x in the carrier of W2
proof
thus x in the carrier of W1 implies
x in the carrier of W2
proof assume A2: x in the carrier of W1;
the carrier of W1 c= the carrier of V by Def2;
then reconsider v = x as Vector of V by A2;
v in W1 by A2,RLVECT_1:def 1;
then v in W2 by A1;
hence thesis by RLVECT_1:def 1;
end;
assume A3: x in the carrier of W2;
the carrier of W2 c= the carrier of V by Def2;
then reconsider v = x as Vector of V by A3;
v in W2 by A3,RLVECT_1:def 1;
then v in W1 by A1;
hence thesis by RLVECT_1:def 1;
end;
then the carrier of W1 = the carrier of W2 by TARSKI:2;
hence thesis by Th37;
end;

theorem
for V being strict RightMod of R, W being strict Submodule of V
holds the carrier of W = the carrier of V
implies W = V
proof let V be strict RightMod of R, W be strict Submodule of V;
assume A1: the carrier of W = the carrier of V;
V is Submodule of V by Th32;
hence thesis by A1,Th37;
end;

theorem
for V being strict RightMod of R, W being strict Submodule of V
holds (for v being Vector of V holds v in W) implies W = V
proof let V be strict RightMod of R, W be strict Submodule of V;
assume for v being Vector of V holds v in W;
then A1: for v be Vector of V holds ( v in W iff v in V) by RLVECT_1:3;
V is Submodule of V by Th32;
hence thesis by A1,Th38;
end;

theorem
the carrier of W = V1 implies V1 is lineary-closed by Lm1;

theorem Th42:
V1 <> {} & V1 is lineary-closed implies
(ex W being strict Submodule of V st V1 = the carrier of W)
proof assume that A1: V1 <> {} and A2: V1 is lineary-closed;
set VV = the carrier of V;
reconsider D = V1 as non empty set by A1;
set A = (the add of V) | [:D,D:];
A3: V1 c= the carrier of V & dom(the add of V) = [:VV,VV:] by FUNCT_2:def 1;
[:D,D:] c= [:VV,VV:] by ZFMISC_1:119;
then A4: dom A = [:D,D:] by A3,RELAT_1:91;
rng A c= D
proof let x;
assume x in rng A;
then consider y such that A5: y in dom A and A6: A.y = x by FUNCT_1:def
5;
consider y1,y2 such that A7: [y1,y2] = y by A4,A5,ZFMISC_1:102;
A8: y1 in D & y2 in D & D c= the carrier of V
by A4,A5,A7,ZFMISC_1:106;
then reconsider y1,y2 as Vector of V;
x = (the add of V).[y1,y2] by A5,A6,A7,FUNCT_1:70
.= (the add of V).(y1,y2) by BINOP_1:def 1
.= y1 + y2 by RLVECT_1:5;
hence thesis by A2,A8,Def1;
end;
then reconsider A as BinOp of D by A4,FUNCT_2:def 1,RELSET_1:11;
set C = (comp V) | D;
V1 c= the carrier of V & dom(comp V) = VV by FUNCT_2:def 1;
then A9: dom C = D by RELAT_1:91;
rng C c= D
proof let x;
assume x in rng C;
then consider y such that A10: y in dom C and A11: C.y = x by FUNCT_1:
def 5;
reconsider y as Vector of V by A9,A10;
x = (comp V).y by A10,A11,FUNCT_1:70
.= - y by VECTSP_1:def 25;
hence thesis by A2,A9,A10,Th5;
end;
then reconsider C as UnOp of D by A9,FUNCT_2:def 1,RELSET_1:11;
set M = (the rmult of V) | [:D,the carrier of R:];
reconsider d = 0.V as Element of D by A2,Th4;
A12: dom(the rmult of V) = [:VV,the carrier of R:] by FUNCT_2:def 1;
[:D,the carrier of R:] c= [:VV,the carrier of R:] by ZFMISC_1:119;
then A13: dom M = [:D,the carrier of R:] by A12,RELAT_1:91;
rng M c= D
proof let x;
assume x in rng M;
then consider y such that A14: y in dom M and A15: M.y = x by FUNCT_1:
def 5;
consider y2,y1 such that A16: [y2,y1] = y by A13,A14,ZFMISC_1:102;
A17: y1 in the carrier of R & y2 in V1 by A13,A14,A16,ZFMISC_1:106;
reconsider y1 as Scalar of R by A13,A14,A16,ZFMISC_1:106;
reconsider y2 as Vector of V by A17;
x = (the rmult of V).[y2,y1] by A14,A15,A16,FUNCT_1:70
.= (the rmult of V).(y2,y1) by BINOP_1:def 1
.= y2 * y1 by VECTSP_2:def 15;
hence thesis by A2,A17,Def1;
end;
then reconsider M as Function of
[:D,the carrier of R:], D by A13,FUNCT_2:def 1,RELSET_1:11;
set W = RightModStr (# D,A,d,M #);
A18: now let a,b be Element of W;
let x,y be Vector of V;
assume A19: x = a & b = y;
thus a + b = (the add of W).(a,b) by RLVECT_1:5
.= A.[a,b] by BINOP_1:def 1
.= (the add of V).[a,b] by A4,FUNCT_1:70
.= (the add of V).(x,y) by A19,BINOP_1:def 1
.= x + y by RLVECT_1:5;
end;
A20: W is Abelian add-associative right_zeroed right_complementable
proof
thus W is Abelian
proof
let a,b be Element of W;
reconsider x = a, y = b as Vector of V by TARSKI:def 3;
thus a + b = y + x by A18
.= b + a by A18;
end;
hereby let a,b,c be Element of W;
reconsider x = a, y = b, z = c as Vector of V by TARSKI:def 3;
A21: a + b = x + y & b + c = y + z by A18;
hence a + b + c = x + y + z by A18
.= x + (y + z) by RLVECT_1:def 6
.= a + (b + c) by A18,A21;
end;
A22: 0.W = 0.V by RLVECT_1:def 2;
hereby let a be Element of W;
reconsider x = a as Vector of V by TARSKI:def 3;
thus a + 0.W = x + 0.V by A18,A22
.= a by VECTSP_1:7;
end;
let a be Element of W;
reconsider x = a as Vector of V by TARSKI:def 3;
reconsider b' = (comp V).x as Vector of V;
C.x in D by FUNCT_2:7;
then reconsider b = ((comp V)|D).x as Element of W;
A23:     b = b' by FUNCT_1:72;
take b;
thus a + b = x + b' by A18,A23
.= x + -x by VECTSP_1:def 25
.= 0.W by A22,RLVECT_1:16;
end;
W is RightMod-like
proof let a,b be Scalar of R;
let v,w be Vector of W;
reconsider x = v, y = w as Vector of V by TARSKI:def 3;
A24: now let a be Scalar of R;
let x be Element of W;
let y be Vector of V;
assume A25: y = x;
A26: [x,a] in dom M by A13;
thus x * a = M.(y,a) by A25,VECTSP_2:def 15
.= M.[y,a] by BINOP_1:def 1
.= (the rmult of V).[y,a] by A25,A26,FUNCT_1:70
.= (the rmult of V).(y,a) by BINOP_1:def 1
.= y * a by VECTSP_2:def 15;
end;
then A27: v * b = x * b & v * a = x * a & w * a = y * a;
then v * a + w * a = x * a + y * a & v + w = x + y by A18;
hence (v + w) * a = (x + y) * a by A24
.= x * a + y * a by VECTSP_2:def 23
.= v * a + w * a by A18,A27;
thus v * (a + b) = x * (a + b) by A24
.= x * a + x * b by VECTSP_2:def 23
.= v * a + v * b by A18,A27;
thus v * (b * a) = x * (b * a) by A24
.= x * b * a by VECTSP_2:def 23
.= v * b * a by A24,A27;
thus v * 1_ R = x * 1_ R by A24
.= v by VECTSP_2:def 23;
end;
then reconsider W as RightMod of R by A20;
the Zero of W = the Zero of V by RLVECT_1:def 2;
then reconsider W as strict Submodule of V by Def2;
take W;
thus thesis;
end;

definition let R; let V;
func (0).V -> strict Submodule of V means :Def3:
the carrier of it = {0.V};
correctness
proof {0.V} is lineary-closed & {0.V} <> {} by Th7;
hence thesis by Th37,Th42;
end;
end;

definition let R; let V;
func (Omega).V -> strict Submodule of V equals
:Def4: the RightModStr of V;
coherence
proof set W = the RightModStr of V;
A1:     now let a; let v,w be Vector of W, v',w' be Vector of V such that
A2:        v=v' & w=w';
thus v+w = (the add of W).(v,w) by RLVECT_1:5
.= v'+w' by A2,RLVECT_1:5;
thus v*a = (the rmult of W).(v,a) by VECTSP_2:def 15
.= v'*a by A2,VECTSP_2:def 15;
end;
A3:  W is Abelian add-associative right_zeroed right_complementable
proof
thus W is Abelian
proof
let x,y be Element of W;
reconsider x' = x, y' = y as Vector of V;
thus x+y = y'+x' by A1
.= y+x by A1;
end;
A4:     0.W = the Zero of W by RLVECT_1:def 2
.= 0.V by RLVECT_1:def 2;
hereby let x,y,z be Element of W;
reconsider x' = x, y' = y, z' = z as Vector of V;
A5:     x + y = x' + y' & y + z = y' + z' by A1;
hence (x+y)+z = (x'+y')+z' by A1
.= x'+(y'+z') by RLVECT_1:def 6
.= x+(y+z) by A1,A5;
end;
hereby let x be Element of W;
reconsider x' = x as Vector of V;
thus x+(0.W) = x'+0.V by A1,A4
.= x by RLVECT_1:10;
end;
let x be Element of W;
reconsider x' = x as Vector of V;
consider b being Vector of V such that
A6:     x' + b = 0.V by RLVECT_1:def 8;
reconsider b' = b as Element of W;
take b';
thus x+b' = 0.W by A1,A4,A6;
end;
W is RightMod-like
proof
let x,y be Element of R,
v,w be Element of W;
reconsider v' = v, w' = w as Vector of V;
A7:     v'*x = v*x & v'*y = v*y & w'*x = w*x by A1;
v+w = v'+w' by A1;
hence (v+w)*x = (v'+w')*x by A1
.= v'*x+w'*x by VECTSP_2:def 23
.= v*x+w*x by A1,A7;
thus v*(x+y) = v'*(x+y) by A1
.= v'*x+v'*y by VECTSP_2:def 23
.= v*x +v*y by A1,A7;
thus v*(y*x) = v'*(y*x) by A1
.= (v'*y)*x by VECTSP_2:def 23
.= (v*y)*x by A1,A7;
thus v*(1_ R) = v'*(1_ R) by A1
.= v by VECTSP_2:def 23;
end;
then reconsider W as RightMod of R by A3;
W is Submodule of V
proof
thus
the carrier of W c= the carrier of V &
the Zero of W = the Zero of V;
thus thesis by FUNCT_2:40;
end;
hence thesis;
end;
end;

canceled 3;

theorem
x in (0).V iff x = 0.V
proof
thus x in (0).V implies x = 0.V
proof assume x in (0).V;
then x in the carrier of (0).V by RLVECT_1:def 1;
then x in {0.V} by Def3;
hence thesis by TARSKI:def 1;
end;
thus thesis by Th25;
end;

theorem Th47:
(0).W = (0).V
proof
the carrier of (0).W = {0.W} &
the carrier of (0).V = {0.V} by Def3;
then the carrier of (0).W =
the carrier of (0).V &
(0).W is Submodule of V by Th19,Th34;
hence thesis by Th37;
end;

theorem Th48:
(0).W1 = (0).W2
proof (0).W1 = (0).V & (0).W2 = (0).V by Th47;
hence thesis;
end;

theorem
(0).W is Submodule of V by Th34;

theorem
(0).V is Submodule of W
proof the carrier of (0).V = {0.V} by Def3
.= {0.W} by Th19
.= {the Zero of W} by RLVECT_1:def 2;
hence thesis by Th35;
end;

theorem
(0).W1 is Submodule of W2
proof (0).W1 = (0).W2 & (0).W2 is Submodule of W2 by Th48;
hence thesis;
end;

canceled;

theorem
for V being strict RightMod of R
holds V is Submodule of (Omega).V
proof let V be strict RightMod of R;
V is Submodule of V by Th32;
hence thesis by Def4;
end;

definition let R; let V; let v,W;
func v + W -> Subset of V equals
:Def5: {v + u : u in W};
coherence
proof
defpred P[set] means ex u st \$1 = v + u & u in W;
consider X being set such that
A1: for x being set holds x in X iff x in the carrier of V &
P[x] from Separation;
X c= the carrier of V
proof let x be set;
assume x in X;
hence x in the carrier of V by A1;
end;
then reconsider X as Subset of V;
set Y = {v + u : u in W};
X = Y
proof
thus X c= Y
proof let x be set;
assume x in X;
then ex u st x = v + u & u in W by A1;
hence thesis;
end;
thus Y c= X
proof let x be set;
assume x in Y;
then ex u st x = v + u & u in W;
hence thesis by A1;
end;
end;
hence thesis;
end;
end;

Lm2: 0.V + W = the carrier of W
proof set A = {0.V + u : u in W};
A1: 0.V + W = A by Def5;
A2: A c= the carrier of W
proof let x be set;
assume x in A;
then consider u such that A3: x = 0.V + u and A4: u in W;
x = u by A3,VECTSP_1:7;
hence thesis by A4,RLVECT_1:def 1;
end;
the carrier of W c= A
proof let x be set;
assume x in the carrier of W;
then A5: x in W by RLVECT_1:def 1;
then x in V by Th17;
then reconsider y = x as Element of V by RLVECT_1:def 1;
0.V + y = x by VECTSP_1:7;
hence thesis by A5;
end;
hence thesis by A1,A2,XBOOLE_0:def 10;
end;

definition let R; let V; let W;
mode Coset of W -> Subset of V means
:Def6: ex v st it = v + W;
existence
proof
reconsider VW = the carrier of W as Subset of V by Def2;
take VW; take 0.V;
thus thesis by Lm2;
end;
end;

reserve B,C for Coset of W;

canceled 3;

theorem Th57:  x in v + W iff ex u st u in W & x = v + u
proof
thus x in v + W implies ex u st u in W & x = v + u
proof assume x in v + W;
then x in {v + u : u in W} by Def5;
then consider u such that A1: x = v + u & u in W;
take u;
thus thesis by A1;
end;
given u such that A2: u in W & x = v + u;
x in {v + v1 : v1 in W} by A2;
hence thesis by Def5;
end;

theorem Th58:
0.V in v + W iff v in W
proof set A = {v + u : u in W};
thus 0.V in v + W implies v in W
proof assume 0.V in v + W;
then 0.V in A by Def5;
then consider u such that A1: 0.V = v + u and A2: u in W;
v = - u by A1,VECTSP_1:63;
hence thesis by A2,Th30;
end;
assume v in W;
then A3: - v in W by Th30;
0.V = v + (- v) by VECTSP_1:66;
then 0.V in A by A3;
hence thesis by Def5;
end;

theorem Th59:
v in v + W
proof v + 0.V = v & 0.V in W by Th25,VECTSP_1:7;
then v in {v + u : u in W};
hence thesis by Def5;
end;

theorem
0.V + W = the carrier of W by Lm2;

theorem Th61:
v + (0).V = {v}
proof set A = {v + u : u in (0).V};
thus v + (0).V c= {v}
proof let x be set;
assume x in v + (0).V;
then x in A by Def5;
then consider u such that A1: x = v + u and A2: u in (0).V;
the carrier of (0).V = {0.V} &
u in the carrier of (0).V by A2,Def3,RLVECT_1:def 1;
then u = 0.V by TARSKI:def 1;
then x = v by A1,VECTSP_1:7;
hence thesis by TARSKI:def 1;
end;
let x be set;
assume x in {v};
then A3: x = v by TARSKI:def 1;
0.V in (0).V & v = v + 0.V by Th25,VECTSP_1:7;
then x in A by A3;
hence thesis by Def5;
end;

Lm3: v in W iff v + W = the carrier of W
proof set A = {v + u : u in W};
thus v in W implies v + W = the carrier of W
proof assume A1: v in W;
thus v + W c= the carrier of W
proof let x be set;
assume x in v + W;
then x in A by Def5;
then consider u such that A2: x = v + u and A3: u in W;
v + u in W by A1,A3,Th28;
hence thesis by A2,RLVECT_1:def 1;
end;
let x be set;
assume x in the carrier of W;
then reconsider y = x, z = v as
Element of W by A1,RLVECT_1:def 1;
reconsider y1 = y, z1 = z as Vector of V by Th18;
A4: y - z in W by RLVECT_1:def 1;
A5: z + (y - z) = z + (y + (- z)) by RLVECT_1:def 11
.= y + (z + (- z)) by RLVECT_1:def 6
.= y + 0.W by VECTSP_1:66
.= x by VECTSP_1:7;
A6: y - z = y1 - z1 by Th24;
A7: y1 - z1 in W by A4,Th24;
z1 + (y1 - z1) = x by A5,A6,Th21;
then x in A by A7;
hence thesis by Def5;
end;
assume A8: v + W = the carrier of W;
assume A9: not v in W;
0.V in W & v + 0.V = v by Th25,VECTSP_1:7;
then v in {v + u : u in W};
then v in the carrier of W by A8,Def5;
hence thesis by A9,RLVECT_1:def 1;
end;

theorem Th62:
v + (Omega).V = the carrier of V
proof
A1: the carrier of the RightModStr of V = the carrier of V;
(Omega).V = the RightModStr of V by Def4;
then v in (Omega).V by RLVECT_1:3;
then v + (Omega).V = the carrier of (Omega).V by Lm3;
hence thesis by A1,Def4;
end;

theorem Th63:
0.V in v + W iff v + W = the carrier of W
proof
(0.V in v + W iff v in W) &
(v in W iff v + W = the carrier of W) by Lm3,Th58;
hence thesis;
end;

theorem
v in W iff v + W = the carrier of W by Lm3;

theorem
v in W implies (v * a) + W = the carrier of W
proof set A = {v * a + u : u in W};
assume A1: v in W;
thus (v * a) + W c= the carrier of W
proof let x be set;
assume x in (v * a) + W;
then x in A by Def5;
then consider u such that A2: x = v * a + u and A3: u in W;
v * a in W by A1,Th29;
then v * a + u in W by A3,Th28;
hence thesis by A2,RLVECT_1:def 1;
end;
let x be set;
assume A4: x in the carrier of W;
the carrier of W c= the carrier of V &
v in V by Def2,RLVECT_1:3;
then reconsider y = x as
Element of V by A4;
v * a in W & x in W by A1,A4,Th29,RLVECT_1:def 1;
then A5: y - v * a in W by Th31;
v * a + (y - v * a) = v * a + (y + - (v * a)) by RLVECT_1:def 11
.= y + (v * a + - (v * a)) by RLVECT_1:def 6
.= y + 0.V by VECTSP_1:66
.= x by VECTSP_1:7;
then x in A by A5;
hence thesis by Def5;
end;

theorem Th66:
u in W iff v + W = (v + u) + W
proof
set A = {v + v1 : v1 in W};
set B = {(v + u) + v2 : v2 in W};
thus u in W implies v + W = (v + u) + W
proof assume A1: u in W;
thus v + W c= (v + u) + W
proof let x be set;
assume x in v + W;
then x in A by Def5;
then consider v1 such that A2: x = v + v1 and A3: v1 in W;
A4: v1 - u in W by A1,A3,Th31;
(v + u) + (v1 - u) = v + (u + (v1 - u)) by RLVECT_1:def 6
.= v + ((v1 + u) - u) by RLVECT_1:42
.= v + (v1 + (u - u)) by RLVECT_1:42
.= v + (v1 + 0.V) by VECTSP_1:66
.= x by A2,VECTSP_1:7;
then x in B by A4;
hence thesis by Def5;
end;
let x be set;
assume x in (v + u) + W;
then x in B by Def5;
then consider v2 such that A5: x = (v + u) + v2 and A6: v2 in W;
A7: u + v2 in W by A1,A6,Th28;
x = v + (u + v2) by A5,RLVECT_1:def 6;
then x in A by A7;
hence thesis by Def5;
end;
assume A8: v + W = (v + u) + W;
0.V in W & v + 0.V = v by Th25,VECTSP_1:7;
then v in A;
then v in (v + u) + W by A8,Def5;
then v in B by Def5;
then consider u1 such that A9: v = (v + u) + u1 and A10: u1 in W;
v = v + 0.V & v = v + (u + u1) by A9,VECTSP_1:7;
then u + u1 = 0.V by RLVECT_1:22;
then u = - u1 by VECTSP_1:63;
hence thesis by A10,Th30;
end;

theorem
u in W iff v + W = (v - u) + W
proof
A1: (- u in W iff v + W = (v + (- u)) + W) & v + (- u) = v - u
by Th66,RLVECT_1:def 11;
- u in W implies u in W
proof assume - u in W;
then - (- u) in W by Th30;
hence thesis by RLVECT_1:30;
end;
hence thesis by A1,Th30;
end;

theorem Th68:
v in u + W iff u + W = v + W
proof set A = {u + v1 : v1 in W}; set B = {v + v2 : v2 in W};
thus v in u + W implies u + W = v + W
proof assume v in u + W;
then v in A by Def5;
then consider z being Vector of V such that A1: v = u + z and
A2: z in W;
thus u + W c= v + W
proof let x be set;
assume x in u + W;
then x in A by Def5;
then consider v1 such that A3: x = u + v1 and A4: v1 in W;
A5: v1 - z in W by A2,A4,Th31;
v - z = u + (z - z) by A1,RLVECT_1:42
.= u + 0.V by VECTSP_1:66
.= u by VECTSP_1:7;
then x = (v + (- z)) + v1 by A3,RLVECT_1:def 11
.= v + (v1 + (- z)) by RLVECT_1:def 6
.= v + (v1 - z) by RLVECT_1:def 11;
then x in B by A5;
hence thesis by Def5;
end;
let x be set;
assume x in v + W;
then x in B by Def5;
then consider v2 such that A6: x = v + v2 and A7: v2 in W;
A8: z + v2 in W by A2,A7,Th28;
x = u + (z + v2) by A1,A6,RLVECT_1:def 6;
then x in A by A8;
hence thesis by Def5;
end;
thus thesis by Th59;
end;

theorem Th69:
u in v1 + W & u in v2 + W implies v1 + W = v2 + W
proof assume that A1: u in v1 + W and A2: u in v2 + W;
set A = {v1 + u1 : u1 in W};
set B = {v2 + u2 : u2 in W};
u in A by A1,Def5;
then consider x1 being Vector of V such that A3: u = v1 + x1 and A4: x1 in
W;
u in B by A2,Def5;
then consider x2 being Vector of V such that A5: u = v2 + x2 and A6: x2 in
W;
thus v1 + W c= v2 + W
proof let x be set;
assume x in v1 + W;
then x in A by Def5;
then consider u1 such that A7: x = v1 + u1 and A8: u1 in W;
u - x1 = v1 + (x1 - x1) by A3,RLVECT_1:42
.= v1 + 0.V by VECTSP_1:66
.= v1 by VECTSP_1:7;
then A9: x = (v2 + (x2 - x1)) + u1 by A5,A7,RLVECT_1:42
.= v2 + ((x2 - x1) + u1) by RLVECT_1:def 6;
x2 - x1 in W by A4,A6,Th31;
then (x2 - x1) + u1 in W by A8,Th28;
then x in B by A9;
hence thesis by Def5;
end;
let x be set;
assume x in v2 + W;
then x in B by Def5;
then consider u1 such that A10: x = v2 + u1 and A11: u1 in W;
u - x2 = v2 + (x2 - x2) by A5,RLVECT_1:42
.= v2 + 0.V by VECTSP_1:66
.= v2 by VECTSP_1:7;
then A12: x = (v1 + (x1 - x2)) + u1 by A3,A10,RLVECT_1:42
.= v1 + ((x1 - x2) + u1) by RLVECT_1:def 6;
x1 - x2 in W by A4,A6,Th31;
then (x1 - x2) + u1 in W by A11,Th28;
then x in A by A12;
hence thesis by Def5;
end;

theorem Th70:
v in W implies v * a in v + W
proof assume A1: v in W;
A2: v * a = v * (a - 0.R) by RLVECT_1:26
.= v * (a - (1_ R - 1_ R)) by RLVECT_1:28
.= v * ((a - 1_ R) + 1_ R) by RLVECT_1:43
.= v * (a - 1_ R) + v * 1_ R by VECTSP_2:def 23
.= v + v * (a - 1_ R) by VECTSP_2:def 23;
v * (a - 1_ R) in W by A1,Th29;
then v * a in {v + u : u in W} by A2;
hence thesis by Def5;
end;

theorem
v in W implies - v in v + W
proof assume v in W;
then v * (- 1_ R) in v + W by Th70;
hence thesis by MOD_1:37;
end;

theorem Th72:
u + v in v + W iff u in W
proof set A = {v + v1 : v1 in W};
thus u + v in v + W implies u in W
proof assume u + v in v + W;
then u + v in A by Def5;
then consider v1 such that A1: u + v = v + v1 and A2: v1 in W;
thus thesis by A1,A2,RLVECT_1:21;
end;
assume u in W;
then u + v in A;
hence thesis by Def5;
end;

theorem
v - u in v + W iff u in W
proof
A1: v - u = (- u) + v by RLVECT_1:def 11;
A2: u in W implies - u in W by Th30;
- u in W implies - (- u) in W by Th30;
hence thesis by A1,A2,Th72,RLVECT_1:30;
end;

canceled;

theorem
u in v + W iff
(ex v1 st v1 in W & u = v - v1)
proof set A = {v + v2 : v2 in W};
thus u in v + W implies (ex v1 st v1 in W & u = v - v1)
proof assume u in v + W;
then u in A by Def5;
then consider v1 such that A1: u = v + v1 and A2: v1 in W;
take x = - v1;
thus x in W by A2,Th30;
u = v + (- (- v1)) by A1,RLVECT_1:30
.= v - (- v1) by RLVECT_1:def 11;
hence thesis;
end;
given v1 such that A3: v1 in W & u = v - v1;
u = v + (- v1) & - v1 in W by A3,Th30,RLVECT_1:def 11;
then u in A;
hence thesis by Def5;
end;

theorem Th76:
(ex v st v1 in v + W & v2 in v + W) iff v1 - v2 in W
proof
thus (ex v st v1 in v + W & v2 in v + W) implies v1 - v2 in W
proof given v such that A1: v1 in v + W and A2: v2 in v + W;
consider u1 such that A3: u1 in W and A4: v1 = v + u1 by A1,Th57;
consider u2 such that A5: u2 in W and A6: v2 = v + u2 by A2,Th57;
v1 - v2 = (u1 + v) + (- (v + u2)) by A4,A6,RLVECT_1:def 11
.= (u1 + v) + ((- v) - u2) by VECTSP_1:64
.= ((u1 + v) + (- v)) - u2 by RLVECT_1:42
.= (u1 + (v + (- v))) - u2 by RLVECT_1:def 6
.= (u1 + 0.V) - u2 by RLVECT_1:16
.= u1 - u2 by VECTSP_1:7;
hence thesis by A3,A5,Th31;
end;
assume v1 - v2 in W;
then A7: - (v1 - v2) in W by Th30;
take v1;
thus v1 in v1 + W by Th59;
v1 + (- (v1 - v2)) = v1 + ((- v1) + v2) by VECTSP_1:64
.= (v1 + (- v1)) + v2 by RLVECT_1:def 6
.= 0.V + v2 by RLVECT_1:16
.= v2 by VECTSP_1:7;
hence thesis by A7,Th57;
end;

theorem Th77:
v + W = u + W implies
(ex v1 st v1 in W & v + v1 = u)
proof
assume A1: v + W = u + W;
take v1 = u - v;
v in u + W by A1,Th59;
then v in {u + u2 : u2 in W} by Def5;
then consider u1 such that A2: v = u + u1 and A3: u1 in W;
0.V = (u + u1) - v by A2,VECTSP_1:66
.= u + (u1 - v) by RLVECT_1:42
.= u + ((- v) + u1) by RLVECT_1:def 11
.= (u + (- v)) + u1 by RLVECT_1:def 6
.= u1 + (u - v) by RLVECT_1:def 11;
then v1 = - u1 by VECTSP_1:63;
hence v1 in W by A3,Th30;
thus v + v1 = (u + v) - v by RLVECT_1:42
.= u + (v - v) by RLVECT_1:42
.= u + 0.V by VECTSP_1:66
.= u by VECTSP_1:7;
end;

theorem Th78:
v + W = u + W implies
(ex v1 st v1 in W & v - v1 = u)
proof
assume A1: v + W = u + W;
take v1 = v - u;
u in v + W by A1,Th59;
then u in {v + u2 : u2 in W} by Def5;
then consider u1 such that A2: u = v + u1 and A3: u1 in W;
0.V = (v + u1) - u by A2,VECTSP_1:66
.= v + (u1 - u) by RLVECT_1:42
.= v + ((- u) + u1) by RLVECT_1:def 11
.= (v + (- u)) + u1 by RLVECT_1:def 6
.= u1 + (v - u) by RLVECT_1:def 11;
then v1 = - u1 by VECTSP_1:63;
hence v1 in W by A3,Th30;
thus v - v1 = (v - v) + u by RLVECT_1:43
.= 0.V + u by VECTSP_1:66
.= u by VECTSP_1:7;
end;

theorem Th79:
for W1,W2 being strict Submodule of V
holds v + W1 = v + W2 iff W1 = W2
proof let W1,W2 be strict Submodule of V;
thus v + W1 = v + W2 implies W1 = W2
proof assume A1: v + W1 = v + W2;
the carrier of W1 = the carrier of W2
proof A2: the carrier of W1 c=
the carrier of V by Def2;
A3: the carrier of W2 c=
the carrier of V by Def2;
thus the carrier of W1 c=
the carrier of W2
proof let x be set;
assume A4: x in the carrier of W1;
then reconsider y = x as
Element of V by A2;
set z = v + y;
x in W1 by A4,RLVECT_1:def 1;
then z in {v + u : u in W1};
then z in v + W2 by A1,Def5;
then z in {v + u : u in W2} by Def5;
then consider u such that A5: z = v + u and A6: u in W2;
y = u by A5,RLVECT_1:21;
hence thesis by A6,RLVECT_1:def 1;
end;
let x be set;
assume A7: x in the carrier of W2;
then reconsider y = x as
Element of V by A3;
set z = v + y;
x in W2 by A7,RLVECT_1:def 1;
then z in {v + u : u in W2};
then z in v + W1 by A1,Def5;
then z in {v + u : u in W1} by Def5;
then consider u such that A8: z = v + u and A9: u in W1;
y = u by A8,RLVECT_1:21;
hence thesis by A9,RLVECT_1:def 1;
end;
hence thesis by Th37;
end;
thus thesis;
end;

theorem Th80:
for W1,W2 being strict Submodule of V
holds v + W1 = u + W2 implies W1 = W2
proof let W1,W2 be strict Submodule of V;
assume A1: v + W1 = u + W2;
set V1 = the carrier of W1;
set V2 = the carrier of W2;
assume A2: W1 <> W2;
then V1 <> V2 by Th37;
then A3: not V1 c= V2 or not V2 c= V1 by XBOOLE_0:def 10;
A4: now assume A5: V1 \ V2 <> {};
consider x being Element of V1 \ V2;
x in V1 & not x in V2 by A5,XBOOLE_0:def 4;
then A6: x in W1 & not x in W2 by RLVECT_1:def 1;
then x in V by Th17;
then reconsider x as Element of V by RLVECT_1:def 1;
set z = v + x;
z in {v + u2 : u2 in W1} by A6;
then z in u + W2 by A1,Def5;
then z in {u + u2 : u2 in W2} by Def5;
then consider u1 such that A7: z = u + u1 and A8: u1 in W2;
x = 0.V + x by VECTSP_1:7
.= (v + (- v)) + x by VECTSP_1:66
.= - v + (u + u1) by A7,RLVECT_1:def 6;
then A9: (v + (- v + (u + u1))) + W1 = v + W1 by A6,Th66;
v + (- v + (u + u1)) = (v + (- v)) + (u + u1) by RLVECT_1:def 6
.= 0.V + (u + u1) by VECTSP_1:66
.= u + u1 by VECTSP_1:7;
then (u + u1) + W2 = (u + u1) + W1 by A1,A8,A9,Th66;
hence thesis by A2,Th79;
end;
now assume A10: V2 \ V1 <> {};
consider x being Element of V2 \ V1;
x in V2 & not x in V1 by A10,XBOOLE_0:def 4;
then A11: x in W2 & not x in W1 by RLVECT_1:def 1;
then x in V by Th17;
then reconsider x as Element of V by RLVECT_1:def 1;
set z = u + x;
z in {u + u2 : u2 in W2} by A11;
then z in v + W1 by A1,Def5;
then z in {v + u2 : u2 in W1} by Def5;
then consider u1 such that A12: z = v + u1 and A13: u1 in W1;
x = 0.V + x by VECTSP_1:7
.= (u + (- u)) + x by VECTSP_1:66
.= - u + (v + u1) by A12,RLVECT_1:def 6;
then A14: (u + (- u + (v + u1))) + W2 = u + W2 by A11,Th66;
u + (- u + (v + u1)) = (u + (- u)) + (v + u1) by RLVECT_1:def 6
.= 0.V + (v + u1) by VECTSP_1:66
.= v + u1 by VECTSP_1:7;
then (v + u1) + W1 = (v + u1) + W2 by A1,A13,A14,Th66;
hence thesis by A2,Th79;
end;
hence thesis by A3,A4,XBOOLE_1:37;
end;

theorem ex C st v in C
proof
reconsider C = v + W as Coset of W by Def6;
take C;
thus thesis by Th59;
end;

theorem
C is lineary-closed iff C = the carrier of W
proof
thus C is lineary-closed implies C = the carrier of W
proof assume A1: C is lineary-closed;
consider v such that A2: C = v + W by Def6;
C <> {} by A2,Th59;
then 0.V in v + W by A1,A2,Th4;
hence thesis by A2,Th63;
end;
thus thesis by Lm1;
end;

theorem
for W1,W2 being strict Submodule of V
for C1 being Coset of W1, C2 being Coset of W2
holds C1 = C2 implies W1 = W2
proof
let W1,W2 be strict Submodule of V;
let C1 be Coset of W1, C2 be Coset of W2;
A1:    ex v1 being Vector of V st C1 = v1 + W1 by Def6;
ex v2 being Vector of V st C2 = v2 + W2 by Def6;
hence thesis by A1,Th80;
end;

theorem
{v} is Coset of (0).V
proof v + (0).V = {v} by Th61;
hence thesis by Def6;
end;

theorem
V1 is Coset of (0).V implies (ex v st V1 = {v})
proof assume V1 is Coset of (0).V;
then consider v such that A1: V1 = v + (0).V by Def6;
take v;
thus thesis by A1,Th61;
end;

theorem
the carrier of W is Coset of W
proof the carrier of W = 0.V + W by Lm2;
hence thesis by Def6;
end;

theorem
the carrier of V is Coset of (Omega).V
proof
the carrier of V c= the carrier of V;
then reconsider A = the carrier of V as Subset of V;
consider v;
A = v + (Omega).V by Th62;
hence thesis by Def6;
end;

theorem
V1 is Coset of (Omega).V implies V1 = the carrier of V
proof assume V1 is Coset of (Omega).V;
then ex v st V1 = v + (Omega).V by Def6;
hence thesis by Th62;
end;

theorem
0.V in C iff C = the carrier of W
proof
ex v st C = v + W by Def6;
hence thesis by Th63;
end;

theorem Th90:
u in C iff C = u + W
proof
thus u in C implies C = u + W
proof assume A1: u in C;
ex v st C = v + W by Def6;
hence thesis by A1,Th68;
end;
thus thesis by Th59;
end;

theorem
u in C & v in C implies (ex v1 st v1 in W & u + v1 = v)
proof assume u in C & v in C;
then C = u + W & C = v + W by Th90;
hence thesis by Th77;
end;

theorem
u in C & v in C implies (ex v1 st v1 in W & u - v1 = v)
proof assume u in C & v in C;
then C = u + W & C = v + W by Th90;
hence thesis by Th78;
end;

theorem
(ex C st v1 in C & v2 in C) iff v1 - v2 in W
proof
thus (ex C st v1 in C & v2 in C) implies v1 - v2 in W
proof given C such that A1: v1 in C & v2 in C;
ex v st C = v + W by Def6;
hence thesis by A1,Th76;
end;
assume v1 - v2 in W;
then consider v such that A2: v1 in v + W & v2 in v + W by Th76;
reconsider C = v + W as Coset of W by Def6;
take C;
thus thesis by A2;
end;

theorem
u in B & u in C implies B = C
proof assume A1: u in B & u in C;
A2:    ex v1 st B = v1 + W by Def6;
ex v2 st C = v2 + W by Def6;
hence thesis by A1,A2,Th69;
end;