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 Th15:
  phi is increasing implies phi|A is increasing
proof
  assume
A1: for A,B st A in B & B in dom phi holds phi.A in phi.B;
  let B,C such that
A2: B in C and
A3: C in dom (phi|A);
A4: phi.B = (phi|A).B by A2,A3,FUNCT_1:47,ORDINAL1:10;
  dom (phi|A) c= dom phi by RELAT_1:60;
  then phi.B in phi.C by A1,A2,A3;
  hence thesis by A3,A4,FUNCT_1:47;
end;
