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

theorem
  z1-z=z2-z implies z1=z2
proof
A1: Rea (z1-z) = Rea z1 -Rea z & Im1(z1-z)=Im1 z1-Im1 z by QUATERNI:42;
A2: Im2 (z1-z) = Im2 z1 -Im2 z & Im3(z1-z)= Im3 z1- Im3 z by QUATERNI:42;
A3: Im2 (z2-z) = Im2 z2 - Im2 z & Im3(z2-z)= Im3 z2-Im3 z by QUATERNI:42;
A4: Rea (z2-z) = Rea z2 -Rea z & Im1(z2-z)=Im1 z2-Im1 z by QUATERNI:42;
  assume z1-z = z2-z;
  hence thesis by A1,A2,A4,A3,QUATERNI:25;
end;
