theorem Th31:
  i in dom D1 & D1 <= D2 implies ex j st j in dom D2 & D1.i=D2.j
proof
  assume i in dom D1; then
A1: D1.i in rng D1 by FUNCT_1:def 3;
  assume D1 <= D2;
  then rng D1 c= rng D2;
  then consider x1 being object such that
A2: x1 in dom D2 and
A3: D1.i=D2.x1 by A1,FUNCT_1:def 3;
  reconsider x1 as Element of NAT by A2;
  take x1;
  thus thesis by A2,A3;
end;
