reserve W,X,Y,Z for set,
  f,g for Function,
  a,x,y,z for set;
reserve u,v for Element of Tarski-Class(X),
  A,B,C for Ordinal,
  L for Sequence;
reserve n for Element of omega;
reserve e,u for set;

theorem Th75:
  for F,G be Function st F,G are_fiberwise_equipotent holds rng F = rng G
proof
  let F,G be Function;
  assume
A1: F,G are_fiberwise_equipotent;
  thus rng F c= rng G
  proof
    let x be object;
    assume that
A2: x in rng F and
A3: not x in rng G;
A4: card Coim(F,x) = card Coim(G,x) by A1;
A5: ex y being object st y in dom F & F.y = x by A2,FUNCT_1:def 3;
 Coim(G,x) = {} by A3,Lm3;
  then x in {x} & F"{x} = {} by A4,CARD_1:5,26,TARSKI:def 1;
    hence contradiction by A5,FUNCT_1:def 7;
  end;
  let x be object;
  assume that
A6: x in rng G and
A7: not x in rng F;
A8: card Coim(G,x) = card Coim(F,x) by A1;
A9: ex y being object st y in dom G & G.y = x by A6,FUNCT_1:def 3;
 Coim(F,x) = {} by A7,Lm3;
then  x in {x} & Coim(G,x) = {} by A8,CARD_1:5,26,TARSKI:def 1;
  hence contradiction by A9,FUNCT_1:def 7;
end;
