
theorem Th34:
  for I be non empty set,
      F,G be Group-Family of I,
      x,y be Function
  st for i be Element of I
     ex hi be Homomorphism of F.i,G.i
     st y.i = hi.(x.i)
  holds support(y,G) c= support(x,F)
  proof
    let I be non empty set,
        F,G be Group-Family of I,
        x,y be Function;
    assume
    A1: for i be Element of I holds
        ex hi be Homomorphism of F.i,G.i
        st y.i = hi.(x.i);
    for i be Element of I holds i in support(y,G) implies i in support(x,F)
    proof
      let i be Element of I;
      assume
      A2: i in support(y,G);
      consider hi be Homomorphism of F.i,G.i such that
      A3: y.i = hi.(x.i) by A1;
      ex Z be Group st Z = G.i & hi.(x.i) <> 1_Z & i in I by A2,A3,Def1; then
      x.i <> 1_ F.i by GROUP_6:31;
      hence thesis by Def1;
    end;
    hence thesis;
  end;
