let D be non empty set ; :: thesis: for i being Nat
for T1, T2 being Element of i -tuples_on 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 Nat; :: thesis: for T1, T2 being Element of i -tuples_on 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 Element of i -tuples_on 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

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, Lm2;
hence (F * (id D),u) .: T1,T2 = F .: T1,(u * T2) by FINSEQ_2:87; :: 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 Lm5;
hence (F * (id D),u) .: T1,T2 = F .: T1,(u * T2) by Th88; :: thesis: verum

end;

end;

end;

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