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 Th34:
  Gen w,y implies for u,v,v1 being VECTOR of V holds 2*PProJ(w,y,u
  ,v#v1) = PProJ(w,y,u,v)+PProJ(w,y,u,v1)
proof
  assume
A1: Gen w,y;
  let u,v,v1 be VECTOR of V;
  set a1=pr1(w,y,u),a2=pr2(w,y,u),b1=pr1(w,y,v),b2=pr2(w,y,v), c1=pr1(w,y,v1),
  c2=pr2(w,y,v1);
  thus 2*PProJ(w,y,u,v#v1) = a1*(2*pr1(w,y,v#v1))+ 2*(a2*pr2(w,y,v#v1))
    .= a1*(b1+c1)+ a2*(2*pr2(w,y,v#v1)) by A1,Lm18
    .= a1*(b1+c1)+ a2*(b2+c2) by A1,Lm18
    .= PProJ(w,y,u,v)+PProJ(w,y,u,v1);
end;
