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 Th31:
  Gen w,y implies for u,v being VECTOR of V, a being Real holds
PProJ(w,y,a*u,v)=a*PProJ(w,y,u,v) & PProJ(w,y,u,a*v)=a*PProJ(w,y,u,v) & PProJ(w
  ,y,a*u,v)=PProJ(w,y,u,v)*a & PProJ(w,y,u,a*v)=PProJ(w,y,u,v)*a
proof
  assume
A1: Gen w,y;
A2: now
    let u,v be VECTOR of V,a be Real;
    PProJ(w,y,a*u,v) = (a*pr1(w,y,u))*pr1(w,y,v)+pr2(w,y,a*u)*pr2(w,y,v)
    by A1,Lm17
      .= (a*pr1(w,y,u))*pr1(w,y,v)+(a*pr2(w,y,u))*pr2(w,y,v) by A1,Lm17
      .= a*PProJ(w,y,u,v);
    hence PProJ(w,y,a*u,v) = a*PProJ(w,y,u,v);
  end;
  let u,v be VECTOR of V,a be Real;
  thus PProJ(w,y,a*u,v) = a*PProJ(w,y,u,v) by A2;
  thus PProJ(w,y,u,a*v) = PProJ(w,y,a*v,u) .= a*PProJ(w,y,v,u) by A2
    .= a*PProJ(w,y,u,v);
  thus PProJ(w,y,a*u,v) = PProJ(w,y,u,v)*a by A2;
  thus PProJ(w,y,u,a*v) = PProJ(w,y,a*v,u) .= a*PProJ(w,y,v,u) by A2
    .= PProJ(w,y,u,v)*a;
end;
