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 Th59:
  |.z.| = 0 implies z = 0
proof
  assume
A1: |.z.| = 0;
A2: 0 <= (Rea z)^2 by XREAL_1:63;
A3: 0 <= (Im1 z)^2 by XREAL_1:63;
A4: 0 <= (Im2 z)^2 by XREAL_1:63;
  0 <= (Im3 z)^2 by XREAL_1:63; then
A5: 0 = (Rea z)^2+(Im1 z)^2+ (Im2 z)^2 + (Im3 z)^2
    by A1,A2,A3,A4,SQUARE_1:25; then
A6: Rea z = 0 by Lm9;
A7: Im1 z = 0 by A5,Lm9;
A8: Im2 z = 0 by A5,Lm9;
  Im3 z = 0 by A5,Lm9;
  hence thesis by A6,A7,A8,Lm6,Th17;
end;
