theorem Th19:
  C <> {} & (for A st A in dom fi holds fi.A = A*^C) implies fi is increasing
proof
  assume that
A1: C <> {} and
A2: for A st A in dom fi holds fi.A = A*^C;
  let A,B;
  assume that
A3: A in B and
A4: B in dom fi;
A5: fi.B = B*^C by A2,A4;
  fi.A = A*^C by A2,A3,A4,ORDINAL1:10;
  hence thesis by A1,A3,A5,ORDINAL2:40;
end;
