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 Th77:
  for F,f,g being Function st [x,y] in dom(F*(f,g)) holds
    (F*(f,g)).(x,y) = F.(f.x,g.y)
proof
  let F,f,g be Function such that
A1: [x,y] in dom(F*(f,g));
  [x,y] in dom [:f,g:] by A1,FUNCT_1:11;
  then [x,y] in [:dom f,dom g:] by FUNCT_3:def 8;
  then [:f,g:].(x,y) = [f.x,g.y] by FUNCT_3:65;
  hence thesis by A1,FUNCT_1:12;
end;
