### Several Properties of Fields. Field Theory

by
Jozef Bialas

Copyright (c) 1990 Association of Mizar Users

MML identifier: REALSET3
[ MML identifier index ]

```environ

vocabulary VECTSP_1, REALSET2, FUNCT_1, BOOLE, BINOP_1, REALSET3;
notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, FUNCT_1, FUNCT_2, REALSET2,
BINOP_1;
constructors REALSET2, XBOOLE_0;
clusters REALSET1, REALSET2, RELSET_1, ZFMISC_1, XBOOLE_0;
requirements SUBSET, BOOLE;
theorems FUNCT_2, ZFMISC_1, REALSET2, TARSKI, XBOOLE_0;
schemes FUNCT_2;

begin

:: Properties of fields

theorem Th1:
for F being Field holds compf(F).(ndf(F)) = ndf(F)
proof
let F be Field;
odf(F).[ndf(F),ndf(F)] = ndf(F) by REALSET2:13;
hence thesis by REALSET2:26;
end;

theorem Th2:
for F being Field holds revf(F).(nmf(F)) = nmf(F)
proof
let F be Field;
omf(F).[nmf(F),nmf(F)] = nmf(F) by REALSET2:22;
hence thesis by REALSET2:42;
end;

theorem Th3:
for F being Field holds
for a,b being Element of suppf(F) holds
compf(F).(odf(F).[a,compf(F).b]) = odf(F).[b,compf(F).a]
proof
let F be Field;
let a,b be Element of suppf(F);
A1:odf(F).[a,compf(F).b] is Element of suppf(F) by REALSET2:28;
A2:odf(F).[b,compf(F).a] is Element of suppf(F) by REALSET2:28;
then odf(F).[compf(F).b,odf(F).[b,compf(F).a]] =
odf(F).[odf(F).[b,compf(F).a],compf(F).b] by REALSET2:12
.= odf(F).[odf(F).[compf(F).a,b],compf(F).b] by REALSET2:12
.= odf(F).[compf(F).a,odf(F).[b,compf(F).b]] by REALSET2:11
.= odf(F).[compf(F).a,ndf(F)] by REALSET2:def 8
.= compf(F).a by REALSET2:13;
then odf(F).[odf(F).[a,compf(F).b],odf(F).[b,compf(F).a]] =
odf(F).[a,compf(F).a] by A2,REALSET2:11
.= ndf(F) by REALSET2:def 8;
hence thesis by A1,A2,REALSET2:26;
end;

theorem Th4:
for F being Field holds
for a,b being Element of suppf(F)\{ndf(F)} holds
revf(F).(omf(F).[a,revf(F).b]) = omf(F).[b,revf(F).a]
proof
let F be Field;
let a,b be Element of suppf(F)\{ndf(F)};
A1:omf(F).[a,revf(F).b] is Element of suppf(F)\{ndf(F)} by REALSET2:44;
A2:omf(F).[b,revf(F).a] is Element of suppf(F)\{ndf(F)} by REALSET2:44;
then omf(F).[revf(F).b,omf(F).[b,revf(F).a]] =
omf(F).[omf(F).[b,revf(F).a],revf(F).b] by REALSET2:21
.= omf(F).[omf(F).[revf(F).a,b],revf(F).b] by REALSET2:21
.= omf(F).[revf(F).a,omf(F).[b,revf(F).b]] by REALSET2:20
.= omf(F).[revf(F).a,nmf(F)] by REALSET2:def 9
.= revf(F).a by REALSET2:22;
then omf(F).[omf(F).[a,revf(F).b],omf(F).[b,revf(F).a]] =omf(F).[a,revf(F).a]
by A2,REALSET2:20
.= nmf(F) by REALSET2:def 9;
hence thesis by A1,A2,REALSET2:42;
end;

theorem Th5:
for F being Field holds
for a,b being Element of suppf(F) holds
compf(F).(odf(F).[a,b]) = odf(F).[compf(F).a,compf(F).b]
proof
let F be Field;
let a,b be Element of suppf(F);
thus compf(F).(odf(F).[a,b]) = compf(F).(odf(F).[a,compf(F).(compf(F).b)])
by REALSET2:27
.= odf(F).[compf(F).b,compf(F).a] by Th3
.= odf(F).[compf(F).a,compf(F).b] by REALSET2:12;
end;

theorem Th6:
for F being Field,
a,b being Element of suppf(F)\{ndf(F)} holds
revf(F).(omf(F).[a,b]) = omf(F).[revf(F).a,revf(F).b]
proof
let F be Field;
let a,b be Element of suppf(F)\{ndf(F)};
thus revf(F).(omf(F).[a,b]) = revf(F).(omf(F).[a,revf(F).(revf(F).b)]) by
REALSET2:43
.= omf(F).[revf(F).b,revf(F).a] by Th4
.= omf(F).[revf(F).a,revf(F).b] by REALSET2:21;
end;

theorem Th7:
for F being Field holds
for a,b,c,d being Element of suppf(F) holds
odf(F).[a,compf(F).b] = odf(F).[c,compf(F).d] iff
odf(F).[a,d] = odf(F).[b,c]
proof
let F be Field;
let a,b,c,d be Element of suppf(F);
A1:odf(F).[c,compf(F).d] is Element of suppf(F) by REALSET2:28;
A2:odf(F).[a,compf(F).b] = odf(F).[c,compf(F).d] implies
odf(F).[a,d] = odf(F).[b,c]
proof
assume
A3:odf(F).[a,compf(F).b] = odf(F).[c,compf(F).d];
A4:a = odf(F).[a,ndf(F)] by REALSET2:13
.= odf(F).[a,odf(F).[b,compf(F).b]] by REALSET2:def 8
.= odf(F).[a,odf(F).[compf(F).b,b]] by REALSET2:12
.= odf(F).[odf(F).[c,compf(F).d],b] by A3,REALSET2:11
.= odf(F).[b,odf(F).[c,compf(F).d]] by A1,REALSET2:12;
c = odf(F).[c,ndf(F)] by REALSET2:13
.= odf(F).[c,odf(F).[d,compf(F).d]] by REALSET2:def 8
.= odf(F).[c,odf(F).[compf(F).d,d]] by REALSET2:12
.= odf(F).[odf(F).[c,compf(F).d],d] by REALSET2:11;
hence thesis by A1,A4,REALSET2:11;
end;
odf(F).[a,d] = odf(F).[b,c] implies
odf(F).[a,compf(F).b] = odf(F).[c,compf(F).d]
proof
assume
A5:odf(F).[a,d] = odf(F).[b,c];
a = odf(F).[a,ndf(F)] by REALSET2:13
.= odf(F).[a,odf(F).[d,compf(F).d]] by REALSET2:def 8
.= odf(F).[odf(F).[b,c],compf(F).d] by A5,REALSET2:11
.= odf(F).[b,odf(F).[c,compf(F).d]] by REALSET2:11
.= odf(F).[odf(F).[c,compf(F).d],b] by A1,REALSET2:12;
hence odf(F).[a,compf(F).b] =
odf(F).[odf(F).[c,compf(F).d],odf(F).[b,compf(F).b]]
by A1,REALSET2:11
.= odf(F).[odf(F).[c,compf(F).d],ndf(F)] by REALSET2:def 8
.= odf(F).[c,compf(F).d] by A1,REALSET2:13;
end;
hence thesis by A2;
end;

theorem Th8:
for F being Field holds
for a,c being Element of suppf(F) holds
for b,d being Element of suppf(F)\{ndf(F)} holds
omf(F).[a,revf(F).b] = omf(F).[c,revf(F).d] iff omf(F).[a,d] = omf(F).[b,c]
proof
let F be Field;
let a,c be Element of suppf(F);
let b,d be Element of suppf(F)\{ndf(F)};
A1:d is Element of suppf(F) & b is Element of suppf(F) by XBOOLE_0:def 4;
A2:revf(F).d is Element of suppf(F) &
revf(F).b is Element of suppf(F) by XBOOLE_0:def 4;
then A3:omf(F).[c,revf(F).d] is Element of suppf(F) by REALSET2:28;
A4:omf(F).[a,revf(F).b] = omf(F).[c,revf(F).d] implies
omf(F).[a,d] = omf(F).[b,c]
proof
assume
A5:omf(F).[a,revf(F).b] = omf(F).[c,revf(F).d];
A6:a = omf(F).[a,nmf(F)] by REALSET2:39
.= omf(F).[a,omf(F).[b,revf(F).b]] by REALSET2:def 9
.= omf(F).[a,omf(F).[revf(F).b,b]] by A1,A2,REALSET2:38
.= omf(F).[omf(F).[c,revf(F).d],b] by A1,A2,A5,REALSET2:37
.= omf(F).[b,omf(F).[c,revf(F).d]] by A1,A3,REALSET2:38;
c = omf(F).[c,nmf(F)] by REALSET2:39
.= omf(F).[c,omf(F).[d,revf(F).d]] by REALSET2:def 9
.= omf(F).[c,omf(F).[revf(F).d,d]] by A1,A2,REALSET2:38
.= omf(F).[omf(F).[c,revf(F).d],d] by A1,A2,REALSET2:37;
hence thesis by A1,A3,A6,REALSET2:37;
end;
omf(F).[a,d] = omf(F).[b,c] implies
omf(F).[a,revf(F).b] = omf(F).[c,revf(F).d]
proof
assume
A7:omf(F).[a,d] = omf(F).[b,c];
a = omf(F).[a,nmf(F)] by REALSET2:39
.= omf(F).[a,omf(F).[d,revf(F).d]] by REALSET2:def 9
.= omf(F).[omf(F).[b,c],revf(F).d] by A1,A2,A7,REALSET2:37
.= omf(F).[b,omf(F).[c,revf(F).d]] by A1,A2,REALSET2:37
.= omf(F).[omf(F).[c,revf(F).d],b] by A1,A3,REALSET2:38;
hence omf(F).[a,revf(F).b] =
omf(F).[omf(F).[c,revf(F).d],omf(F).[b,revf(F).b]]
by A1,A2,A3,REALSET2:37
.= omf(F).[omf(F).[c,revf(F).d],nmf(F)] by REALSET2:def 9
.= omf(F).[c,revf(F).d] by A3,REALSET2:39;
end;
hence thesis by A4;
end;

theorem
for F being Field holds
for a,b being Element of suppf(F) holds
omf(F).[a,b] = ndf(F) iff (a = ndf(F) or b = ndf(F))
proof
let F be Field;
let a,b be Element of suppf(F);
omf(F).[a,b] = ndf(F) implies (a = ndf(F) or b = ndf(F))
proof
assume
A1:omf(F).[a,b] = ndf(F);
assume a <> ndf(F);
then not a in {ndf(F)} by TARSKI:def 1;
then A2:a is Element of suppf(F)\{ndf(F)} by XBOOLE_0:def 4;
then revf(F).a is Element of suppf(F)\{ndf(F)} by REALSET2:44;
then A3:revf(F).a is Element of suppf(F) by XBOOLE_0:def 4;
thus b = omf(F).[b,nmf(F)] by REALSET2:39
.= omf(F).[b,omf(F).[a,revf(F).a]] by A2,REALSET2:def 9
.= omf(F).[omf(F).[b,a],revf(F).a] by A3,REALSET2:37
.= omf(F).[ndf(F),revf(F).a] by A1,REALSET2:38
.= ndf(F) by A3,REALSET2:32;
end;
hence thesis by REALSET2:31,32;
end;

theorem
for F being Field holds
for a,b being Element of suppf(F) holds
for c,d being Element of suppf(F)\{ndf(F)} holds
omf(F).[omf(F).[a,revf(F).c],omf(F).[b,revf(F).d]] =
omf(F).[omf(F).[a,b],revf(F).(omf(F).[c,d])]
proof
let F be Field;
let a,b be Element of suppf(F);
let c,d be Element of suppf(F)\{ndf(F)};
A1:revf(F).c is Element of suppf(F) by XBOOLE_0:def 4;
A2:revf(F).d is Element of suppf(F) by XBOOLE_0:def 4;
then A3:omf(F).[b,revf(F).d] is Element of suppf(F) by REALSET2:36;
omf(F).[c,d] is Element of suppf(F)\{ndf(F)} by REALSET2:44;
then revf(F).(omf(F).[c,d])
is Element of suppf(F)\{ndf(F)} by REALSET2:44;
then A4:revf(F).(omf(F).[c,d]) is Element of suppf(F) by XBOOLE_0:def 4;
thus omf(F).[omf(F).[a,revf(F).c],omf(F).[b,revf(F).d]] =
omf(F).[a,omf(F).[revf(F).c,omf(F).[b,revf(F).d]]]
by A1,A3,REALSET2:37
.= omf(F).[a,omf(F).[omf(F).[revf(F).c,b],revf(F).d]]
by A1,A2,REALSET2:37
.= omf(F).[a,omf(F).[omf(F).[b,revf(F).c],revf(F).d]] by A1,REALSET2:38
.= omf(F).[a,omf(F).[b,omf(F).[revf(F).c,revf(F).d]]]
by A1,A2,REALSET2:37
.= omf(F).[a,omf(F).[b,revf(F).(omf(F).[c,d])]] by Th6
.= omf(F).[omf(F).[a,b],revf(F).(omf(F).[c,d])] by A4,REALSET2:37;
end;

theorem
for F being Field holds
for a,b being Element of suppf(F) holds
for c,d being Element of suppf(F)\{ndf(F)} holds
odf(F).[omf(F).[a,revf(F).c],omf(F).[b,revf(F).d]] =
omf(F).[odf(F).[omf(F).[a,d],omf(F).[b,c]],revf(F).(omf(F).[c,d])]
proof
let F be Field;
let a,b be Element of suppf(F);
let c,d be Element of suppf(F)\{ndf(F)};
A1:revf(F).c is Element of suppf(F) by XBOOLE_0:def 4;
A2:revf(F).d is Element of suppf(F) by XBOOLE_0:def 4;
omf(F).[c,d] is Element of suppf(F)\{ndf(F)} by REALSET2:44;
then revf(F).(omf(F).[c,d])
is Element of suppf(F)\{ndf(F)} by REALSET2:44;
then A3:revf(F).(omf(F).[c,d]) is Element of suppf(F) by XBOOLE_0:def 4;
A4:c is Element of suppf(F) by XBOOLE_0:def 4;
A5:d is Element of suppf(F) by XBOOLE_0:def 4;
then A6:omf(F).[a,d] is Element of suppf(F) by REALSET2:36;
A7:omf(F).[b,c] is Element of suppf(F) by A4,REALSET2:36;
A8:a = omf(F).[a,nmf(F)] by REALSET2:39
.= omf(F).[a,omf(F).[d,revf(F).d]] by REALSET2:def 9
.= omf(F).[omf(F).[a,d],revf(F).d] by A2,A5,REALSET2:37;
A9:b = omf(F).[b,nmf(F)] by REALSET2:39
.= omf(F).[b,omf(F).[c,revf(F).c]] by REALSET2:def 9
.= omf(F).[omf(F).[b,c],revf(F).c] by A1,A4,REALSET2:37;
A10:omf(F).[a,revf(F).c] =omf(F).[omf(F).[a,d],omf(F).[revf(F).d,revf(F).c]]
by A1,A2,A6,A8,REALSET2:37
.= omf(F).[omf(F).[a,d],omf(F).[revf(F).c,revf(F).d]]
by A1,A2,REALSET2:38
.= omf(F).[omf(F).[a,d],revf(F).(omf(F).[c,d])] by Th6;
omf(F).[b,revf(F).d] =omf(F).[omf(F).[b,c],omf(F).[revf(F).c,revf(F).d]]
by A1,A2,A7,A9,REALSET2:37
.= omf(F).[omf(F).[b,c],revf(F).(omf(F).[c,d])] by Th6;
hence thesis by A3,A6,A7,A10,REALSET2:10;
end;

definition ::subtraction
let F be Field;
func osf(F) -> BinOp of suppf(F) means
:Def1: for x,y being Element of suppf(F) holds
it.[x,y] = odf(F).[x,compf(F).y];
existence
proof
defpred P[Element of suppf(F),Element of suppf(F),set]
means \$3=odf(F).[\$1,compf(F).\$2];

now let x,y be Element of suppf(F);
reconsider z = odf(F).[x,compf(F).y] as Element of suppf(F)
by REALSET2:28;
take z;
thus z = odf(F).[x,compf(F).y];
end; then
A1: for x being Element of suppf(F) for y being Element of suppf(F)
ex z being Element of suppf(F) st P[x,y,z];
ex f being Function of [:suppf(F),suppf(F):],suppf(F)
st for x being Element of suppf(F) for y being Element of suppf(F) holds
P[x,y,f.[x,y]] from FuncEx2D(A1);
then consider S being BinOp of suppf(F) such that
A2:for x,y being Element of suppf(F) holds
S.[x,y]=odf(F).[x,compf(F).y];
take S;
thus thesis by A2;
end;
uniqueness
proof
let S1,S2 be BinOp of suppf(F) such that
A3:for x,y being Element of suppf(F) holds
S1.[x,y] = odf(F).[x,compf(F).y] and
A4:for x,y being Element of suppf(F) holds
S2.[x,y] = odf(F).[x,compf(F).y];
now let p be set;
assume p in [:suppf(F),suppf(F):];
then consider x,y being set such that
A5:x in suppf(F) & y in suppf(F) & p=[x,y] by ZFMISC_1:def 2;
thus S1.p = odf(F).[x,compf(F).y] by A3,A5
.= S2.p by A4,A5;
end;
hence thesis by FUNCT_2:18;
end;
end;

canceled 2;

theorem
for F being Field holds
for x being Element of suppf(F) holds osf(F).[x,x] = ndf(F)
proof
let F be Field;
let x be Element of suppf(F);
thus osf(F).[x,x] = odf(F).[x,compf(F).x] by Def1
.= ndf(F) by REALSET2:def 8;
end;

theorem Th15:
for F being Field holds
for a,b,c being Element of suppf(F) holds
omf(F).[a,osf(F).[b,c]] = osf(F).[omf(F).[a,b],omf(F).[a,c]]
proof
let F be Field;
let a,b,c be Element of suppf(F);
A1:omf(F).[a,b] is Element of suppf(F) by REALSET2:28;
A2:omf(F).[a,c] is Element of suppf(F) by REALSET2:28;
thus omf(F).[a,osf(F).[b,c]] = omf(F).[a,odf(F).[b,compf(F).c]] by Def1
.= odf(F).[omf(F).[a,b],compf(F).(omf(F).[a,c])] by REALSET2:29
.= osf(F).[omf(F).[a,b],omf(F).[a,c]] by A1,A2,Def1;
end;

theorem Th16:
for F being Field holds
for a,b being Element of suppf(F) holds
osf(F).[a,b] is Element of suppf(F)
proof
let F be Field;
let a,b be Element of suppf(F);
osf(F).[a,b] = odf(F).[a,compf(F).b] by Def1;
hence thesis by REALSET2:28;
end;

theorem
for F being Field holds
for a,b,c being Element of suppf(F) holds
omf(F).[osf(F).[a,b],c] =
osf(F).[omf(F).[a,c],omf(F).[b,c]]
proof
let F be Field;
let a,b,c be Element of suppf(F);
A1:omf(F).[c,b] = omf(F).[b,c] by REALSET2:38;
osf(F).[a,b] is Element of suppf(F) by Th16;
hence omf(F).[osf(F).[a,b],c] = omf(F).[c,osf(F).[a,b]] by REALSET2:38
.= osf(F).[omf(F).[c,a],omf(F).[c,b]] by Th15
.= osf(F).[omf(F).[a,c],omf(F).[b,c]] by A1,REALSET2:38;
end;

theorem
for F being Field holds
for a,b being Element of suppf(F) holds
osf(F).[a,b] = compf(F).(osf(F).[b,a])
proof
let F be Field;
let a,b be Element of suppf(F);
osf(F).[a,b] = odf(F).[a,compf(F).b] by Def1
.= compf(F).(odf(F).[b,compf(F).a]) by Th3;
hence thesis by Def1;
end;

theorem
for F being Field holds
for a,b being Element of suppf(F) holds
osf(F).[compf(F).a,b] = compf(F).(odf(F).[a,b])
proof
let F be Field;
let a,b be Element of suppf(F);
thus osf(F).[compf(F).a,b] = odf(F).[compf(F).a,compf(F).b] by Def1
.= compf(F).(odf(F).[a,b]) by Th5;
end;

theorem Th20:
for F being Field holds
for a,b,c,d being Element of suppf(F) holds
osf(F).[a,b] = osf(F).[c,d] iff odf(F).[a,d] = odf(F).[b,c]
proof
let F be Field;
let a,b,c,d be Element of suppf(F);
A1:osf(F).[a,b] = odf(F).[a,compf(F).b] by Def1;
osf(F).[c,d] = odf(F).[c,compf(F).d] by Def1;
hence thesis by A1,Th7;
end;

theorem
for F being Field holds
for a being Element of suppf(F) holds
osf(F).[ndf(F),a] = compf(F).a
proof
let F be Field;
let a be Element of suppf(F);
thus osf(F).[ndf(F),a] = odf(F).[ndf(F),compf(F).a] by Def1
.= compf(F).a by REALSET2:13;
end;

theorem Th22:
for F being Field holds
for a being Element of suppf(F) holds
osf(F).[a,ndf(F)] = a
proof
let F be Field;
let a be Element of suppf(F);
thus osf(F).[a,ndf(F)] = odf(F).[a,compf(F).ndf(F)] by Def1
.= odf(F).[a,ndf(F)] by Th1
.= a by REALSET2:13;
end;

theorem
for F being Field holds
for a,b,c being Element of suppf(F) holds
odf(F).[a,b] = c iff osf(F).[c,a] = b
proof
let F be Field;
let a,b,c be Element of suppf(F);
set d=ndf(F);
A1:odf(F).[c,d] = c by REALSET2:13;
osf(F).[b,d] = b by Th22;
hence thesis by A1,Th20;
end;

theorem
for F being Field holds
for a,b,c being Element of suppf(F) holds
odf(F).[a,b] = c iff osf(F).[c,b] = a
proof
let F be Field;
let a,b,c be Element of suppf(F);
set d=ndf(F);
A1:odf(F).[c,d] = c by REALSET2:13;
A2:osf(F).[a,d] = a by Th22;
odf(F).[b,a] = odf(F).[a,b] by REALSET2:12;
hence thesis by A1,A2,Th20;
end;

theorem
for F being Field holds
for a,b,c being Element of suppf(F) holds
osf(F).[a,osf(F).[b,c]] = odf(F).[osf(F).[a,b],c]
proof
let F be Field;
let a,b,c be Element of suppf(F);
A1:odf(F).[b,compf(F).c] is Element of suppf(F) by REALSET2:28;
thus osf(F).[a,osf(F).[b,c]] = osf(F).[a,odf(F).[b,compf(F).c]] by Def1
.= odf(F).[a,compf(F).(odf(F).[b,compf(F).c])] by A1,Def1
.= odf(F).[a,odf(F).[compf(F).b,compf(F).(compf(F).c)]] by Th5
.= odf(F).[a,odf(F).[compf(F).b,c]] by REALSET2:27
.= odf(F).[odf(F).[a,compf(F).b],c] by REALSET2:11
.= odf(F).[osf(F).[a,b],c] by Def1;
end;

theorem
for F being Field holds
for a,b,c being Element of suppf(F) holds
osf(F).[a,odf(F).[b,c]] = osf(F).[osf(F).[a,b],c]
proof
let F be Field;
let a,b,c be Element of suppf(F);
A1:odf(F).[b,c] is Element of suppf(F) by REALSET2:28;
A2:osf(F).[a,b] is Element of suppf(F) by Th16;
thus osf(F).[a,odf(F).[b,c]] = odf(F).[a,compf(F).(odf(F).[b,c])]
by A1,Def1
.= odf(F).[a,odf(F).[compf(F).b,compf(F).c]] by Th5
.= odf(F).[odf(F).[a,compf(F).b],compf(F).c] by REALSET2:11
.= odf(F).[osf(F).[a,b],compf(F).c] by Def1
.= osf(F).[osf(F).[a,b],c] by A2,Def1;
end;

definition ::division.
let F be Field;
func ovf(F) -> Function of [:suppf(F),suppf(F)\{ndf(F)}:],suppf(F) means
:Def2: for x being Element of suppf(F),
y being Element of suppf(F)\{ndf(F)} holds
it.[x,y] = omf(F).[x,revf(F).y];
existence
proof
defpred P[Element of suppf(F),Element of suppf(F)\{ndf(F)},set]
means \$3=omf(F).[\$1,revf(F).\$2];
now let x be Element of suppf(F),
y be Element of suppf(F)\{ndf(F)};
revf(F).y is Element of suppf(F) by XBOOLE_0:def 4;
then reconsider z = omf(F).[x,revf(F).y] as Element of suppf(F)
by REALSET2:28;
take z;
thus z = omf(F).[x,revf(F).y];
end; then
A1:  for x being Element of suppf(F) for y being Element of suppf(F)\{ndf(F)}
ex z being Element of suppf(F) st P[x,y,z];
ex f being Function of [:suppf(F),suppf(F)\{ndf(F)}:],suppf(F)
st for x being Element of suppf(F),
y being Element of suppf(F)\{ndf(F)} holds P[x,y,f.[x,y]]
from FuncEx2D(A1);
hence thesis;
end;
uniqueness
proof
let S1,S2 be Function of [:suppf(F),suppf(F)\{ndf(F)}:],suppf(F)
such that
A2:for x being Element of suppf(F),
y being Element of suppf(F)\{ndf(F)} holds
S1.[x,y] = omf(F).[x,revf(F).y] and
A3:for x being Element of suppf(F),
y being Element of suppf(F)\{ndf(F)} holds
S2.[x,y] = omf(F).[x,revf(F).y];
now let p be set;
assume p in [:suppf(F),suppf(F)\{ndf(F)}:];
then consider x,y being set such that
A4:x in suppf(F) & y in suppf(F)\{ndf(F)} & p=[x,y] by ZFMISC_1:def 2;
S1.p = omf(F).[x,revf(F).y] by A2,A4
.= S2.p by A3,A4;
hence S1.p = S2.p;
end;
hence thesis by FUNCT_2:18;
end;
end;

canceled 2;

theorem Th29:
for F being Field holds
for x being Element of suppf(F)\{ndf(F)} holds
ovf(F).[x,x] = nmf(F)
proof
let F be Field;
let x be Element of suppf(F)\{ndf(F)};
x is Element of suppf(F) by XBOOLE_0:def 4;
hence ovf(F).[x,x] = omf(F).[x,revf(F).x] by Def2
.=nmf(F) by REALSET2:def 9;
end;

theorem Th30:
for F being Field holds
for a being Element of suppf(F) holds
for b being Element of suppf(F)\{ndf(F)} holds
ovf(F).[a,b] is Element of suppf(F)
proof
let F be Field;
let a be Element of suppf(F);
let b be Element of suppf(F)\{ndf(F)};
A1:ovf(F).[a,b] = omf(F).[a,revf(F).b] by Def2;
revf(F).b is Element of suppf(F) by XBOOLE_0:def 4;
hence thesis by A1,REALSET2:28;
end;

theorem Th31:
for F being Field holds
for a,b being Element of suppf(F) holds
for c being Element of suppf(F)\{ndf(F)} holds
omf(F).[a,ovf(F).[b,c]] =
ovf(F).[omf(F).[a,b],c]
proof
let F be Field;
let a,b be Element of suppf(F);
let c be Element of suppf(F)\{ndf(F)};
A1:omf(F).[a,b] is Element of suppf(F) by REALSET2:28;
A2:revf(F).c is Element of suppf(F) by XBOOLE_0:def 4;
thus omf(F).[a,ovf(F).[b,c]] = omf(F).[a,omf(F).[b,revf(F).c]] by Def2
.= omf(F).[omf(F).[a,b],revf(F).c] by A2,REALSET2:37
.= ovf(F).[omf(F).[a,b],c] by A1,Def2;
end;

theorem
for F being Field holds
for a being Element of suppf(F)\{ndf(F)} holds
omf(F).[a,ovf(F).[nmf(F),a]] = nmf(F) &
omf(F).[ovf(F).[nmf(F),a],a] = nmf(F)
proof
let F be Field;
let a be Element of suppf(F)\{ndf(F)};
A1:a is Element of suppf(F) by XBOOLE_0:def 4;
nmf(F) in suppf(F)\{ndf(F)};
then A2:nmf(F) is Element of suppf(F) by XBOOLE_0:def 4;
then A3:omf(F).[a,ovf(F).[nmf(F),a]] = ovf(F).[omf(F).[a,nmf(F)],a] by A1,Th31
.= ovf(F).[a,a] by A1,REALSET2:39
.= nmf(F) by Th29;
ovf(F).[nmf(F),a] is Element of suppf(F) by A2,Th30;
hence thesis by A1,A3,REALSET2:38;
end;

canceled 2;

theorem
for F being Field holds
for a,b being Element of suppf(F)\{ndf(F)} holds
ovf(F).[a,b] = revf(F).(ovf(F).[b,a])
proof
let F be Field;
let a,b be Element of suppf(F)\{ndf(F)};
A1:a is Element of suppf(F) & b is Element of suppf(F) by XBOOLE_0:def 4;
then ovf(F).[a,b] = omf(F).[a,revf(F).b] by Def2
.= revf(F).(omf(F).[b,revf(F).a]) by Th4;
hence thesis by A1,Def2;
end;

theorem
for F being Field holds
for a,b being Element of suppf(F)\{ndf(F)} holds
ovf(F).[revf(F).a,b] = revf(F).(omf(F).[a,b])
proof
let F be Field;
let a,b be Element of suppf(F)\{ndf(F)};
revf(F).a is Element of suppf(F) by XBOOLE_0:def 4;
hence ovf(F).[revf(F).a,b] = omf(F).[revf(F).a,revf(F).b] by Def2
.= revf(F).(omf(F).[a,b]) by Th6;
end;

theorem Th37:
for F being Field holds
for a,c being Element of suppf(F) holds
for b,d being Element of suppf(F)\{ndf(F)} holds
ovf(F).[a,b] = ovf(F).[c,d] iff omf(F).[a,d] = omf(F).[b,c]
proof
let F be Field;
let a,c be Element of suppf(F);
let b,d be Element of suppf(F)\{ndf(F)};
A1:ovf(F).[a,b] = omf(F).[a,revf(F).b] by Def2;
ovf(F).[c,d] = omf(F).[c,revf(F).d] by Def2;
hence thesis by A1,Th8;
end;

theorem
for F being Field holds
for a being Element of suppf(F)\{ndf(F)} holds
ovf(F).[nmf(F),a] = revf(F).a
proof
let F be Field;
let a be Element of suppf(F)\{ndf(F)};
nmf(F) in suppf(F)\{ndf(F)};
then nmf(F) is Element of suppf(F) by XBOOLE_0:def 4;
hence ovf(F).[nmf(F),a] = omf(F).[nmf(F),revf(F).a] by Def2
.= revf(F).a by REALSET2:22;
end;

theorem Th39:
for F being Field holds
for a being Element of suppf(F) holds
ovf(F).[a,nmf(F)] = a
proof
let F be Field;
let a be Element of suppf(F);
thus ovf(F).[a,nmf(F)] = omf(F).[a,revf(F).nmf(F)] by Def2
.= omf(F).[a,nmf(F)] by Th2
.= a by REALSET2:39;
end;

theorem
for F being Field holds
for a being Element of suppf(F)\{ndf(F)} holds
for b,c being Element of suppf(F) holds
omf(F).[a,b] = c iff ovf(F).[c,a] = b
proof
let F be Field;
let a be Element of suppf(F)\{ndf(F)};
let b,c be Element of suppf(F);
set d=nmf(F);
A1:omf(F).[c,d] = c by REALSET2:39;
ovf(F).[b,d] = b by Th39;
hence thesis by A1,Th37;
end;

theorem
for F being Field holds
for a,c being Element of suppf(F) holds
for b being Element of suppf(F)\{ndf(F)} holds
omf(F).[a,b] = c iff ovf(F).[c,b] = a
proof
let F be Field;
let a,c be Element of suppf(F);
let b be Element of suppf(F)\{ndf(F)};
set d=nmf(F);
A1:omf(F).[c,d] = c by REALSET2:39;
A2:ovf(F).[a,d] = a by Th39;
b is Element of suppf(F) by XBOOLE_0:def 4;
then omf(F).[b,a] = omf(F).[a,b] by REALSET2:38;
hence thesis by A1,A2,Th37;
end;

theorem
for F being Field holds
for a being Element of suppf(F) holds
for b,c being Element of suppf(F)\{ndf(F)} holds
ovf(F).[a,ovf(F).[b,c]] = omf(F).[ovf(F).[a,b],c]
proof
let F be Field;
let a be Element of suppf(F);
let b,c be Element of suppf(F)\{ndf(F)};
A1:b is Element of suppf(F) & c is Element of suppf(F) by XBOOLE_0:def 4;
A2:omf(F).[b,revf(F).c] is Element of suppf(F)\{ndf(F)} by REALSET2:44;
A3:revf(F).b is Element of suppf(F) by XBOOLE_0:def 4;
thus ovf(F).[a,ovf(F).[b,c]] = ovf(F).[a,omf(F).[b,revf(F).c]] by A1,Def2
.= omf(F).[a,revf(F).(omf(F).[b,revf(F).c])] by A2,Def2
.= omf(F).[a,omf(F).[revf(F).b,revf(F).(revf(F).c)]] by Th6
.= omf(F).[a,omf(F).[revf(F).b,c]] by REALSET2:43
.= omf(F).[omf(F).[a,revf(F).b],c] by A1,A3,REALSET2:37
.= omf(F).[ovf(F).[a,b],c] by Def2;
end;

theorem
for F being Field holds
for a being Element of suppf(F) holds
for b,c being Element of suppf(F)\{ndf(F)} holds
ovf(F).[a,omf(F).[b,c]] = ovf(F).[ovf(F).[a,b],c]
proof
let F be Field;
let a be Element of suppf(F);
let b,c be Element of suppf(F)\{ndf(F)};
A1:revf(F).b is Element of suppf(F) &
revf(F).c is Element of suppf(F) by XBOOLE_0:def 4;
A2:omf(F).[b,c] is Element of suppf(F)\{ndf(F)} by REALSET2:44;
A3:ovf(F).[a,b] is Element of suppf(F) by Th30;
thus ovf(F).[a,omf(F).[b,c]] = omf(F).[a,revf(F).(omf(F).[b,c])] by A2,Def2
.= omf(F).[a,omf(F).[revf(F).b,revf(F).c]] by Th6
.= omf(F).[omf(F).[a,revf(F).b],revf(F).c] by A1,REALSET2:37
.= omf(F).[ovf(F).[a,b],revf(F).c] by Def2
.= ovf(F).[ovf(F).[a,b],c] by A3,Def2;
end;
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