let X, Y be non empty set ; :: thesis: for F being BinOp of Y st F is commutative & F is associative & F is idempotent holds
for Z being non empty set
for G being BinOp of Z st G is commutative & G is associative & G is idempotent holds
for f being Function of X,Y
for g being Function of Y,Z st ( for x, y being Element of Y holds g . (F . x,y) = G . (g . x),(g . y) ) holds
for B being Element of Fin X st B <> {} holds
g . (F $$ B,f) = G $$ B,(g * f)
let F be BinOp of Y; :: thesis: ( F is commutative & F is associative & F is idempotent implies for Z being non empty set
for G being BinOp of Z st G is commutative & G is associative & G is idempotent holds
for f being Function of X,Y
for g being Function of Y,Z st ( for x, y being Element of Y holds g . (F . x,y) = G . (g . x),(g . y) ) holds
for B being Element of Fin X st B <> {} holds
g . (F $$ B,f) = G $$ B,(g * f) )
assume A1: ( F is commutative & F is associative & F is idempotent ) ; :: thesis: for Z being non empty set
for G being BinOp of Z st G is commutative & G is associative & G is idempotent holds
for f being Function of X,Y
for g being Function of Y,Z st ( for x, y being Element of Y holds g . (F . x,y) = G . (g . x),(g . y) ) holds
for B being Element of Fin X st B <> {} holds
g . (F $$ B,f) = G $$ B,(g * f)
let Z be non empty set ; :: thesis: for G being BinOp of Z st G is commutative & G is associative & G is idempotent holds
for f being Function of X,Y
for g being Function of Y,Z st ( for x, y being Element of Y holds g . (F . x,y) = G . (g . x),(g . y) ) holds
for B being Element of Fin X st B <> {} holds
g . (F $$ B,f) = G $$ B,(g * f)
let G be BinOp of Z; :: thesis: ( G is commutative & G is associative & G is idempotent implies for f being Function of X,Y
for g being Function of Y,Z st ( for x, y being Element of Y holds g . (F . x,y) = G . (g . x),(g . y) ) holds
for B being Element of Fin X st B <> {} holds
g . (F $$ B,f) = G $$ B,(g * f) )
assume A2: ( G is commutative & G is associative & G is idempotent ) ; :: thesis: for f being Function of X,Y
for g being Function of Y,Z st ( for x, y being Element of Y holds g . (F . x,y) = G . (g . x),(g . y) ) holds
for B being Element of Fin X st B <> {} holds
g . (F $$ B,f) = G $$ B,(g * f)
let f be Function of X,Y; :: thesis: for g being Function of Y,Z st ( for x, y being Element of Y holds g . (F . x,y) = G . (g . x),(g . y) ) holds
for B being Element of Fin X st B <> {} holds
g . (F $$ B,f) = G $$ B,(g * f)
let g be Function of Y,Z; :: thesis: ( ( for x, y being Element of Y holds g . (F . x,y) = G . (g . x),(g . y) ) implies for B being Element of Fin X st B <> {} holds
g . (F $$ B,f) = G $$ B,(g * f) )
assume A3: for x, y being Element of Y holds g . (F . x,y) = G . (g . x),(g . y) ; :: thesis: for B being Element of Fin X st B <> {} holds
g . (F $$ B,f) = G $$ B,(g * f)
defpred S₁[ Element of Fin X] means ( $1 <> {} implies g . (F $$ $1,f) = G $$ $1,(g * f) );
A4: S₁[ {}. X] ;
A5: for B' being Element of Fin X
for b being Element of X st S₁[B'] holds
S₁[B' \/ {.b.}]

proof

let B be Element of Fin X; :: thesis: for b being Element of X st S₁[B] holds
S₁[B \/ {.b.}]
let x be Element of X; :: thesis: ( S₁[B] implies S₁[B \/ {.x.}] )
assume that
A6: ( B <> {} implies g . (F $$ B,f) = G $$ B,(g * f) ) and
B \/ {x} <> {} ; :: thesis: g . (F $$ (B \/ {.x.}),f) = G $$ (B \/ {.x.}),(g * f)

per cases ( B = {} or B <> {} ) ;

suppose A7: B = {} ; :: thesis: g . (F $$ (B \/ {.x.}),f) = G $$ (B \/ {.x.}),(g * f)

hence g . (F $$ (B \/ {.x.}),f) = g . (f . x) by A1, Th26
.= (g * f) . x by FUNCT_2:21
.= G $$ (B \/ {.x.}),(g * f) by A2, A7, Th26 ;
:: thesis: verum

end;

suppose A8: B <> {} ; :: thesis: g . (F $$ (B \/ {.x.}),f) = G $$ (B \/ {.x.}),(g * f)

hence g . (F $$ (B \/ {.x.}),f) = g . (F . (F $$ B,f),(f . x)) by A1, Th29
.= G . (g . (F $$ B,f)),(g . (f . x)) by A3
.= G . (G $$ B,(g * f)),((g * f) . x) by A6, A8, FUNCT_2:21
.= G $$ (B \/ {.x.}),(g * f) by A2, A8, Th29 ;
:: thesis: verum

end;

end;

end;

thus for B being Element of Fin X holds S₁[B] from SETWISEO:sch 4(A4, A5); :: thesis: verum