reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;
reserve f1,f2 for Ordinal-Sequence;

theorem Th11:
  phi is increasing implies phi"A is epsilon-transitive epsilon-connected set
proof
  assume
A1: for A,B st A in B & B in dom phi holds phi.A in phi.B;
  now
    let X;
    assume
A2: X in phi"A;
    then
A3: X in dom phi by FUNCT_1:def 7;
    hence X is Ordinal;
    reconsider B = X as Ordinal by A3;
A4: phi.X in A by A2,FUNCT_1:def 7;
    thus X c= phi"A
    proof
      let x be object;
      assume
A5:   x in X;
      then x in B;
      then reconsider C = x, D = phi.B as Ordinal;
      reconsider E = phi.C as Ordinal;
      E in D by A1,A3,A5;
      then
A6:   phi.x in A by A4,ORDINAL1:10;
      C in dom phi by A3,A5,ORDINAL1:10;
      hence thesis by A6,FUNCT_1:def 7;
    end;
  end;
  hence thesis by ORDINAL1:19;
end;
