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
  Rea(z + z*') = 2*Rea z & Im1(z + z*') = 0 &
  Im2(z + z*') = 0 & Im3(z + z*') = 0
proof
A1: z = [*Rea z, Im1 z, Im2 z, Im3 z*] by Th17;
  z*' = [*Rea z, -Im1 z, -Im2 z, -Im3 z*] by Th36;
  then z + z*' = [*Rea z+Rea z, Im1 z+ -Im1 z, Im2 z+ -Im2 z, Im3 z+ -Im3 z*]
  by A1,Def6
    .=[*2*Rea z,0,0,0*];
  hence thesis by Th16;
end;
