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;

theorem
  (Rea z)^2 + (Im1 z)^2 +(Im2 z)^2 + (Im3 z)^2 = 0 implies z = 0q
proof
  assume
A1: (Rea z)^2 + (Im1 z)^2 +(Im2 z)^2 + (Im3 z)^2 = 0;
  then
A2: Rea z = 0 by Lm9;
A3: Im1 z = 0 by A1,Lm9;
A4: Im2 z = 0 by A1,Lm9;
  Im3 z = 0 by A1,Lm9;
  hence thesis by A2,A3,A4,Th19;
end;
