reserve x,y,z for set;
reserve f,f1,f2,f3 for FinSequence,
  p,p1,p2,p3 for set,
  i,k for Nat;
reserve D for non empty set,
  p,p1,p2,p3 for Element of D,
  f,f1,f2 for FinSequence of D;

theorem Th62:
  x in rng f & p in rng f & x..f >= p..f implies x in rng(f:-p)
proof
  assume that
A1: x in rng f and
A2: p in rng f and
A3: x..f >= p..f;
  per cases by A3,XXREAL_0:1;
  suppose
A4: x..f > p..f;
    rng f c= D by FINSEQ_1:def 4;
    then reconsider q = x as Element of D by A1;
    f:-p = <*p*>^(f/^p..f) by FINSEQ_5:def 2;
    then
A5: rng(f:-p) = rng<*p*> \/ rng(f/^p..f) by FINSEQ_1:31;
    q in rng(f/^(p..f)) by A1,A4,Th57;
    hence thesis by A5,XBOOLE_0:def 3;
  end;
  suppose
    x..f = p..f;
    then x = f.(p..f) by A1,FINSEQ_4:19
      .= p by A2,FINSEQ_4:19;
    hence thesis by Th61;
  end;
end;
