reserve a,b for object, I,J for set;

theorem Lem12:
  for p,q being FinSequence holds p c< q iff len p < len q &
  for i being Nat st i in dom p holds p.i = q.i
  proof
    let p,q be FinSequence;
    hereby
      assume Z0: p c< q;
      hence len p < len q by TREES_1:6;
      p c= q by Z0,XBOOLE_0:def 8;
      hence for i being Nat st i in dom p holds p.i = q.i by FOMODEL0:51;
    end;
    assume Z2: len p < len q;
    then dom p c< dom q by FINSEQ_3:118;
    then
A1: dom p c= dom q by XBOOLE_0:def 8;
    assume for i being Nat st i in dom p holds p.i = q.i;
    then for i being set st i in dom p holds i in dom q & p.i = q.i by A1;
    hence p c< q by Z2,XBOOLE_0:def 8,FOMODEL0:51;
  end;
