let X, Y, D be non empty set ; :: thesis: for f being Function of X,(Fin Y)
for g being Function of (Fin Y),D
for F being BinOp of D st ( for A, B being Element of Fin Y st A misses B holds
F . ((g . A),(g . B)) = g . (A \/ B) ) & F is commutative & F is associative & F is having_a_unity & g . {} = the_unity_wrt F holds
for I being Element of Fin X st ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) holds
( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y )

let f be Function of X,(Fin Y); :: thesis: for g being Function of (Fin Y),D
for F being BinOp of D st ( for A, B being Element of Fin Y st A misses B holds
F . ((g . A),(g . B)) = g . (A \/ B) ) & F is commutative & F is associative & F is having_a_unity & g . {} = the_unity_wrt F holds
for I being Element of Fin X st ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) holds
( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y )

let g be Function of (Fin Y),D; :: thesis: for F being BinOp of D st ( for A, B being Element of Fin Y st A misses B holds
F . ((g . A),(g . B)) = g . (A \/ B) ) & F is commutative & F is associative & F is having_a_unity & g . {} = the_unity_wrt F holds
for I being Element of Fin X st ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) holds
( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y )

let F be BinOp of D; :: thesis: ( ( for A, B being Element of Fin Y st A misses B holds
F . ((g . A),(g . B)) = g . (A \/ B) ) & F is commutative & F is associative & F is having_a_unity & g . {} = the_unity_wrt F implies for I being Element of Fin X st ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) holds
( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y ) )

assume that
A1: for A, B being Element of Fin Y st A misses B holds
F . ((g . A),(g . B)) = g . (A \/ B) and
A2: ( F is commutative & F is associative ) and
A3: F is having_a_unity and
A4: g . {} = the_unity_wrt F ; :: thesis: for I being Element of Fin X st ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) holds
( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y )

defpred S1[ set ] means for I being Element of Fin X st I = \$1 & ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) holds
( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y );
A5: for I being Element of Fin X
for i being Element of X st S1[I] & not i in I holds
S1[I \/ {i}]
proof
let A be Element of Fin X; :: thesis: for i being Element of X st S1[A] & not i in A holds
S1[A \/ {i}]

let a be Element of X; :: thesis: ( S1[A] & not a in A implies S1[A \/ {a}] )
assume that
A6: S1[A] and
A7: not a in A ; :: thesis: S1[A \/ {a}]
let I be Element of Fin X; :: thesis: ( I = A \/ {a} & ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) implies ( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y ) )

assume that
A8: A \/ {a} = I and
A9: for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ; :: thesis: ( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y )
A10: for x, y being object st x in A & y in A & f . x meets f . y holds
x = y
proof
let x, y be object ; :: thesis: ( x in A & y in A & f . x meets f . y implies x = y )
assume that
A11: x in A and
A12: y in A and
A13: f . x meets f . y ; :: thesis: x = y
A c= I by ;
hence x = y by A9, A11, A12, A13; :: thesis: verum
end;
then A14: F \$\$ (A,(g * f)) = F \$\$ ((f .: A),g) by A6;
A15: union (f .: A) is Element of Fin Y by ;
dom f = X by FUNCT_2:def 1;
then Im (f,a) = {(f . a)} by FUNCT_1:59;
then A16: f .: I = (f .: A) \/ {(f . a)} by ;
A17: F \$\$ ((f .: A),g) = g . (union (f .: A)) by ;
dom (g * f) = X by FUNCT_2:def 1;
then A18: g . (f . a) = (g * f) . a by FUNCT_1:12;
per cases ( not f . a is empty or not f . a in f .: A or ( f . a is empty & f . a in f .: A ) ) ;
suppose A19: ( not f . a is empty or not f . a in f .: A ) ; :: thesis: ( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y )
not f . a in f .: A
proof
A20: A c= I by ;
A21: {a} c= I by ;
A22: a in {a} by TARSKI:def 1;
assume A23: f . a in f .: A ; :: thesis: contradiction
then consider x being object such that
x in dom f and
A24: x in A and
A25: f . x = f . a by FUNCT_1:def 6;
f . x meets f . a by ;
hence contradiction by A7, A9, A24, A22, A21, A20; :: thesis: verum
end;
then A26: F \$\$ ((f .: I),g) = F . ((F \$\$ ((f .: A),g)),((g * f) . a)) by ;
A27: f . a c= Y by FINSUB_1:def 5;
union (f .: A) c= Y by ;
then A28: (union (f .: A)) \/ (f . a) c= Y by ;
now :: thesis: for x being set st x in f .: A holds
not x meets f . a
let x be set ; :: thesis: ( x in f .: A implies not x meets f . a )
assume x in f .: A ; :: thesis: not x meets f . a
then A29: ex y being object st
( y in dom f & y in A & f . y = x ) by FUNCT_1:def 6;
A30: a in {a} by TARSKI:def 1;
A31: A c= I by ;
A32: {a} c= I by ;
assume x meets f . a ; :: thesis: contradiction
hence contradiction by A7, A9, A29, A30, A32, A31; :: thesis: verum
end;
then A33: union (f .: A) misses f . a by ZFMISC_1:80;
union (f .: I) = (union (f .: A)) \/ (union {(f . a)}) by
.= (union (f .: A)) \/ (f . a) by ZFMISC_1:25 ;
hence ( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y ) by ; :: thesis: verum
end;
suppose A34: ( f . a is empty & f . a in f .: A ) ; :: thesis: ( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y )
then A35: (f .: A) \/ {(f . a)} = f .: A by ZFMISC_1:40;
F \$\$ (I,(g * f)) = F . ((F \$\$ ((f .: A),g)),()) by A2, A3, A4, A7, A8, A14, A18, A34, SETWOP_2:2
.= F \$\$ ((f .: I),g) by ;
hence ( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y ) by A6, A10, A16, A35; :: thesis: verum
end;
end;
end;
A36: S1[ {}. X]
proof
A37: {} c= Y ;
let I be Element of Fin X; :: thesis: ( I = {}. X & ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) implies ( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y ) )

assume that
A38: {}. X = I and
for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ; :: thesis: ( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y )
A39: f .: I = {}. (Fin Y) by A38;
F \$\$ (I,(g * f)) = g . {} by ;
hence ( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y ) by ; :: thesis: verum
end;
for I being Element of Fin X holds S1[I] from hence for I being Element of Fin X st ( for x, y being object st x in I & y in I & f . x meets f . y holds
x = y ) holds
( F \$\$ (I,(g * f)) = F \$\$ ((f .: I),g) & F \$\$ ((f .: I),g) = g . (union (f .: I)) & union (f .: I) is Element of Fin Y ) ; :: thesis: verum