reserve a,b,c,d,x,y,w,z,x1,x2,x3,x4 , X for set;
reserve A for non empty set;
reserve i,j,k for Element of NAT;
reserve a,b,c,d for Real;
reserve y,r,s,x,t,w for Element of RAT+;
reserve z,z1,z2,z3,z4 for Quaternion;
 reserve x for Real;

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;
