
theorem Th45:
for Y,S be non empty set, F be PartFunc of Y,S, M be
  Function of S,ExtREAL st M is nonnegative holds M*F is nonnegative
proof
   let Y,S be non empty set;
   let F be PartFunc of Y,S;
   let M be Function of S,ExtREAL;
   assume A1: M is nonnegative;

   now let n be object;
    per cases;
    suppose n in dom(M*F); then
     (M*F).n = M.(F.n) by FUNCT_1:12;
     hence (M*F).n >= 0 by A1,SUPINF_2:51;
    end;
    suppose not n in dom(M*F);
     hence (M*F).n >= 0 by FUNCT_1:def 2;
    end;
   end;
   hence thesis by SUPINF_2:51;
end;
