let D be non empty set ; :: thesis: for I, J being non empty set
for F being BinOp of D
for f being Function of [:I,J:],D
for g being Function of I,D
for X being Element of Fin I
for Y being Element of Fin J st ( for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) ) & F is having_a_unity & F is commutative & F is associative holds
F $$ ([:X,Y:],f) = F $$ (X,g)
let I, J be non empty set ; :: thesis: for F being BinOp of D
for f being Function of [:I,J:],D
for g being Function of I,D
for X being Element of Fin I
for Y being Element of Fin J st ( for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) ) & F is having_a_unity & F is commutative & F is associative holds
F $$ ([:X,Y:],f) = F $$ (X,g)
let F be BinOp of D; :: thesis: for f being Function of [:I,J:],D
for g being Function of I,D
for X being Element of Fin I
for Y being Element of Fin J st ( for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) ) & F is having_a_unity & F is commutative & F is associative holds
F $$ ([:X,Y:],f) = F $$ (X,g)
let f be Function of [:I,J:],D; :: thesis: for g being Function of I,D
for X being Element of Fin I
for Y being Element of Fin J st ( for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) ) & F is having_a_unity & F is commutative & F is associative holds
F $$ ([:X,Y:],f) = F $$ (X,g)
let g be Function of I,D; :: thesis: for X being Element of Fin I
for Y being Element of Fin J st ( for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) ) & F is having_a_unity & F is commutative & F is associative holds
F $$ ([:X,Y:],f) = F $$ (X,g)
let X be Element of Fin I; :: thesis: for Y being Element of Fin J st ( for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) ) & F is having_a_unity & F is commutative & F is associative holds
F $$ ([:X,Y:],f) = F $$ (X,g)
let Y be Element of Fin J; :: thesis: ( ( for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) ) & F is having_a_unity & F is commutative & F is associative implies F $$ ([:X,Y:],f) = F $$ (X,g) )
assume that
A1: for i being Element of I holds g . i = F $$ (Y,((curry f) . i)) and
A2: ( F is having_a_unity & F is commutative & F is associative ) ; :: thesis: F $$ ([:X,Y:],f) = F $$ (X,g)
defpred S₁[ Element of Fin I] means F $$ ([:$1,Y:],f) = F $$ ($1,g);
A3: for X1 being Element of Fin I
for x being Element of I st S₁[X1] holds
S₁[X1 \/ {.x.}]

proof

let X1 be Element of Fin I; :: thesis: for x being Element of I st S₁[X1] holds
S₁[X1 \/ {.x.}]
let x be Element of I; :: thesis: ( S₁[X1] implies S₁[X1 \/ {.x.}] )
assume A4: F $$ ([:X1,Y:],f) = F $$ (X1,g) ; :: thesis: S₁[X1 \/ {.x.}]
reconsider s = {.x.} as Element of Fin I ;

per cases ( x in X1 or not x in X1 ) ;

suppose x in X1 ; :: thesis: S₁[X1 \/ {.x.}]

then X1 \/ {x} = X1 by ZFMISC_1:40;
hence S₁[X1 \/ {.x.}] by A4; :: thesis: verum

end;

suppose not x in X1 ; :: thesis: S₁[X1 \/ {.x.}]

then A5: X1 misses {x} by ZFMISC_1:50;
then A6: [:X1,Y:] misses [:s,Y:] by ZFMISC_1:104;
thus F $$ ([:(X1 \/ {.x.}),Y:],f) = F $$ (([:X1,Y:] \/ [:s,Y:]),f) by ZFMISC_1:97
.= F . ((F $$ ([:X1,Y:],f)),(F $$ ([:s,Y:],f))) by A2, A6, SETWOP_2:4
.= F . ((F $$ (X1,g)),(F $$ (s,g))) by A1, A2, A4, Th29
.= F $$ ((X1 \/ {.x.}),g) by A2, A5, SETWOP_2:4 ; :: thesis: verum

end;

end;

end;

A7: S₁[ {}. I]

proof

reconsider T = {}. [:I,J:] as Element of Fin [:I,J:] ;
T = [:({}. I),Y:] by ZFMISC_1:90;
then F $$ ([:({}. I),Y:],f) = the_unity_wrt F by A2, SETWISEO:31;
hence S₁[ {}. I] by A2, SETWISEO:31; :: thesis: verum

end;

for X1 being Element of Fin I holds S₁[X1] from SETWISEO:sch 4(A7, A3);
hence F $$ ([:X,Y:],f) = F $$ (X,g) ; :: thesis: verum