reserve z1,z2,z3,z4,z for Quaternion;

theorem
  z is Real implies z-z3 = [* Rea z -Rea z3,-Im1 z3,-Im2 z3,-Im3 z3*]
proof
  reconsider z1 = z+(-z3) as Quaternion;
  assume
A1: z is Real; then
A2: Im3 z = 0 by Lm1;
  set z2 = [*Rea z- Rea z3,Im1 z -Im1 z3,Im2 z -Im2 z3,Im3 z -Im3 z3*];
A3: Rea z2 = Rea z +(-Rea z3) by QUATERNI:23
    .= Rea z + Rea (-z3) by QUATERNI:41
    .= Rea z1 by QUATERNI:36;
A4: Im1 z2 = Im1 z+(-Im1 z3) by QUATERNI:23
    .=Im1 z+Im1 (- z3) by QUATERNI:41
    .= Im1 z1 by QUATERNI:36;
A5: Im3 z2 = Im3 z +(-Im3 z3) by QUATERNI:23
    .=Im3 z+Im3 (- z3) by QUATERNI:41
    .= Im3 z1 by QUATERNI:36;
A6: Im2 z2 = Im2 z +(-Im2 z3) by QUATERNI:23
    .=Im2 z+Im2 (- z3) by QUATERNI:41
    .= Im2 z1 by QUATERNI:36;
  Im1 z = 0 & Im2 z = 0 by A1,Lm1;
  hence thesis by A2,A3,A4,A6,A5,QUATERNI:25;
end;
