reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;

theorem Th30:
  Gen w,y implies for u,v,v1 holds PProJ(w,y,u,v+v1)=PProJ(w,y,u,v
)+PProJ(w,y,u,v1) & PProJ(w,y,u,v-v1)=PProJ(w,y,u,v)-PProJ(w,y,u,v1) & PProJ(w,
  y,v-v1,u)=PProJ(w,y,v,u)-PProJ(w,y,v1,u) & PProJ(w,y,v+v1,u)=PProJ(w,y,v,u)+
  PProJ(w,y,v1,u)
proof
  assume
A1: Gen w,y;
  let u,v,v1 be VECTOR of V;
A2: now
    let u,v,v1 be VECTOR of V;
A3: PProJ(w,y,u,v-v1) = pr1(w,y,u)*(pr1(w,y,v)-pr1(w,y,v1))+pr2(w,y,u)*pr2
    (w,y,v-v1) by A1,Lm17
      .= pr1(w,y,u)*(pr1(w,y,v)-pr1(w,y,v1))+pr2(w,y,u)*(pr2(w,y,v)-pr2(w,y,
    v1)) by A1,Lm17
      .= PProJ(w,y,u,v)-PProJ(w,y,u,v1);
    PProJ(w,y,u,v+v1) = pr1(w,y,u)*(pr1(w,y,v)+pr1(w,y,v1))+pr2(w,y,u)*pr2
    (w,y,v+v1) by A1,Lm17
      .= pr1(w,y,u)*(pr1(w,y,v)+pr1(w,y,v1))+pr2(w,y,u)*(pr2(w,y,v)+pr2(w,y,
    v1)) by A1,Lm17
      .= PProJ(w,y,u,v)+PProJ(w,y,u,v1);
    hence
    PProJ(w,y,u,v+v1)=PProJ(w,y,u,v)+PProJ(w,y,u,v1) & PProJ(w,y,u,v-v1)=
    PProJ(w,y,u,v)-PProJ(w,y,u,v1) by A3;
  end;
  hence PProJ(w,y,u,v+v1) = PProJ(w,y,u,v)+PProJ(w,y,u,v1) & PProJ(w,y,u,v-v1)
  = PProJ(w,y,u,v)-PProJ(w,y,u,v1);
  thus PProJ(w,y,v-v1,u) = PProJ(w,y,u,v-v1)
    .= PProJ(w,y,u,v)-PProJ (w,y,u,v1) by A2
    .= PProJ(w,y,v,u)-PProJ (w,y,v1,u);
  thus PProJ(w,y,v+v1,u)= PProJ(w,y,u,v+v1)
    .= PProJ(w,y,u,v)+PProJ (w,y,u,v1) by A2
    .= PProJ(w,y,v,u)+PProJ (w,y,v1,u);
end;
