theorem
  Im3 z <= |.z.|
proof
  0<= (Im3 z)^2 by XREAL_1:63; then
  A1: sqrt ((Im3 z)^2) <= sqrt ((Rea z)^2 + (Im1 z)^2 + (Im2 z)^2 + (Im3 z )^2)
  by Lm28,SQUARE_1:26;
  per cases;
  suppose Im3 z >= 0;
    hence thesis by A1,SQUARE_1:22;
  end;
  suppose
A2: Im3 z < 0;
    then -Im3 z <= |.z.| by A1,SQUARE_1:23;
    hence thesis by A2;
  end;
end;
