reserve x,y for set;
reserve C,C9,D,D9,E for non empty set;
reserve c for Element of C;
reserve c9 for Element of C9;
reserve d,d1,d2,d3,d4,e for Element of D;
reserve d9 for Element of D9;
reserve i,j for natural Number;
reserve F for Function of [:D,D9:],E;
reserve p,q for FinSequence of D,
  p9,q9 for FinSequence of D9;
reserve f,f9 for Function of C,D,
  h for Function of D,E;
reserve T,T1,T2,T3 for Tuple of i,D;
reserve T9 for Tuple of i, D9;
reserve S for Tuple of j, D;
reserve S9 for Tuple of j, D9;
reserve F,G for BinOp of D;
reserve u for UnOp of D;
reserve H for BinOp of E;

theorem Th80:
  for F being Function of [:D,D9:],E, f being Function of C,D,
      g being Function of C9,D9 holds (F*(f,g)).(c,c9) = F.(f.c,g.c9)
proof
  let F be Function of [:D,D9:],E, f be Function of C,D, g be Function of C9,
  D9;
  set H = F*(f,g);
  H is Function of [:C,C9:],E by Th78;
  then dom H = [:C,C9:] by FUNCT_2:def 1;
  then [c,c9] in dom H by ZFMISC_1:def 2;
  hence thesis by Th77;
end;
