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 Th65:
  |.-z.| = |.z.|
proof
A1: -z = [*-Rea z, -Im1 z, -Im2 z, -Im3 z*] by Th55;
  then
A2: Rea -z = - Rea z by Th16;
A3: Im1 -z = - Im1 z by A1,Th16;
A4: Im2 -z = - Im2 z by A1,Th16;
  Im3 -z = - Im3 z by A1,Th16;
  hence thesis by A2,A3,A4;
end;
