let D be non empty set ; :: thesis: for i being natural Number
for T1, T2 being Tuple of i,D
for F being BinOp of D
for u being UnOp of D holds (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2))
let i be natural Number ; :: thesis: for T1, T2 being Tuple of i,D
for F being BinOp of D
for u being UnOp of D holds (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2))
let T1, T2 be Tuple of i,D; :: thesis: for F being BinOp of D
for u being UnOp of D holds (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2))
let F be BinOp of D; :: thesis: for u being UnOp of D holds (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2))
let u be UnOp of D; :: thesis: (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2))

now :: thesis: (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2))

per cases ( i = 0 or i <> 0 ) ;

suppose i = 0 ; :: thesis: (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2))

then ( (F * ((id D),u)) .: (T1,T2) = <*> D & u * T2 = <*> D ) by Lm1;
hence (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2)) by FINSEQ_2:73; :: thesis: verum

end;

suppose i <> 0 ; :: thesis: (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2))

then reconsider C = Seg i as non empty set ;
( T1 is Function of C,D & T2 is Function of C,D ) by Lm4;
hence (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2)) by Th82; :: thesis: verum

end;

end;

end;

hence (F * ((id D),u)) .: (T1,T2) = F .: (T1,(u * T2)) ; :: thesis: verum