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
  Gen x,y implies (u,v,u1,v1 are_Ort_wrt x,y iff
  u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y)
proof
  assume
A1: Gen x,y;
A2: now
    assume u=v;
    then v-u=0.V by RLVECT_1:15;
    then v-u,v1-u1 are_Ort_wrt x,y by A1,ANALMETR:5;
    hence u,v,u1,v1 are_Ort_wrt x,y by ANALMETR:def 3;
  end;
  now
    assume
A3: u<>v;
    set u2=Orte(x,y,u),v2=Orte(x,y,v);
A4: v-u<>0.V by A3,RLVECT_1:21;
    u,v,u2,v2 are_Ort_wrt x,y by A1,Th24;
    then
A5: v-u,v2-u2 are_Ort_wrt x,y by ANALMETR:def 3;
A6: now
      assume u,v,u1,v1 are_Ort_wrt x,y;
      then v-u,v1-u1 are_Ort_wrt x,y by ANALMETR:def 3;
      then ex a,b st
      a*(v2-u2)=b*(v1-u1) & (a<>0 or b<>0) by A1,A4,A5,ANALMETR:9;
      then u2,v2 // u1,v1 or u2,v2 // v1,u1 by ANALMETR:14;
      hence u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y;
    end;
    now
      assume u,v,u1,v1 are_COrte_wrt x,y or u,v,v1,u1 are_COrte_wrt x,y;
      then u2,v2 // u1,v1 or u2,v2 // v1,u1;
      then consider a,b such that
A7:   a*(v2-u2)=b*(v1-u1) and
A8:   a<>0 or b<>0 by ANALMETR:14;
A9:   now
        assume
A10:    b=0;
        then 0.V = a*(v2-u2) by A7,RLVECT_1:10;
        then v2-u2=0.V by A8,A10,RLVECT_1:11;
        then v2=u2 by RLVECT_1:21;
        then u=v by A1,Th13;
        then v-u=0.V by RLVECT_1:15;
        then v-u,v1-u1 are_Ort_wrt x,y by A1,ANALMETR:5;
        hence u,v,u1,v1 are_Ort_wrt x,y by ANALMETR:def 3;
      end;
      now
        assume
A11:    b<>0;
        ((b")*a)*(v2-u2)=(b")*(b*(v1-u1)) by A7,RLVECT_1:def 7;
        then ((b")*a)*(v2-u2)=((b")*b)*(v1-u1) by RLVECT_1:def 7;
        then ((b")*a)*(v2-u2)=1*(v1-u1) by A11,XCMPLX_0:def 7;
        then v1-u1=((b")*a)*(v2-u2) by RLVECT_1:def 8;
        then v-u,v1-u1 are_Ort_wrt x,y by A5,ANALMETR:7;
        hence u,v,u1,v1 are_Ort_wrt x,y by ANALMETR:def 3;
      end;
      hence u,v,u1,v1 are_Ort_wrt x,y by A9;
    end;
    hence thesis by A6;
  end;
  hence thesis by A2,Th20;
end;
