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 Th19:
  Rea z = 0 & Im1 z = 0 & Im2 z = 0 & Im3 z = 0 implies z = 0q
proof
  assume that
A1: Rea z = 0 and
A2: Im1 z = 0 and
A3: Im2 z = 0 and
A4: Im3 z = 0;
A5: Rea z = Rea [*0,0,0,0*] by A1,Th16;
A6: Im1 z = Im1 [*0,0,0,0*] by A2,Th16;
A7: Im2 z = Im2 [*0,0,0,0*] by A3,Th16;
  Im3 z = Im3 [*0,0,0,0*] by A4,Th16;
  hence thesis by A5,A6,A7,Lm6,Th18;
end;
