reserve i for Nat,
  j for Element of NAT,
  X,Y,x,y,z for set;

theorem Th6:
  for x,y,z being set st x in {z}* & y in {z}* & card x = card y holds x = y
proof
  let x,y,z be set such that
A1: x in {z}* and
A2: y in {z}* and
A3: card x = card y;
  reconsider x, y as FinSequence of {z} by A1,A2,FINSEQ_1:def 11;
A4: dom x = Seg len x by FINSEQ_1:def 3
    .= dom y by A3,FINSEQ_1:def 3;
  now
    let i be Nat;
    assume
A5: i in dom x;
    then
A6: x .i in rng x by FUNCT_1:def 3;
A7: y.i in rng y by A4,A5,FUNCT_1:def 3;
    thus x .i = z by A6,TARSKI:def 1
      .= y.i by A7,TARSKI:def 1;
  end;
  hence thesis by A4,FINSEQ_1:13;
end;
