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
  z*' = 0 implies z = 0
proof
  assume
A1: z*' = 0;
A2: z*' = [*Rea z, -Im1 z, -Im2 z, -Im3 z*] by Th36;
A3: Rea z*' = 0 by A1,Lm6,Th16;
A4: Im1 z*' = 0 by A1,Lm6,Th16;
A5: Im2 z*'= 0 by A1,Lm6,Th16;
A6: Im3 z*' = 0 by A1,Lm6,Th16;
A7: Rea z*' = Rea z by A2,Th16;
A8: Im1 z*' = -Im1 z by A2,Th16;
A9: Im2 z*'= -Im2 z by A2,Th16;
  Im3 z*' = -Im3 z by A2,Th16;
  hence thesis by A3,A4,A5,A6,A7,A8,A9,Th19;
end;
