reserve V for RealLinearSpace,
  u,u1,u2,v,v1,v2,w,w1,x,y for VECTOR of V,
  a,a1,a2,b,b1,b2,c1,c2,n,k1,k2 for Real;

theorem Th24:
  Gen x,y implies u,v,Orte(x,y,u),Orte(x,y,v) are_Ort_wrt x,y
proof
  assume
A1: Gen x,y;
  set w = Orte(x,y,v) - Orte(x,y,u);
A2: w = Orte(x,y,v-u) by A1,Th11
    .= pr2(x,y,v-u)*x + (-pr1(x,y,v-u))*y;
  PProJ(x,y,v-u,w) = pr1(x,y,v-u)*pr1(x,y,w) + pr2(x,y,v-u)*pr2(x,y,w)
  by GEOMTRAP:def 6
    .= pr1(x,y,v-u)*pr2(x,y,v-u) + pr2(x,y,v-u)*pr2(x,y,w)
  by A1,A2,Lm6
    .= pr1(x,y,v-u)*pr2(x,y,v-u) + (-pr1(x,y,v-u))*pr2(x,y,v-u) by A1,A2,Lm6
    .= 0;
  then v-u,w are_Ort_wrt x,y by A1,GEOMTRAP:32;
  hence thesis by ANALMETR:def 3;
end;
