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 iff ex u2,v2 st u2<>v2 &
  u2,v2,u,v are_COrte_wrt x,y & u2,v2,u1,v1 are_COrte_wrt x,y)
proof
  assume
A1: Gen x,y;
A2: u,v // u1,v1 implies ex u2,v2 st u2<>v2 &
  u2,v2,u,v are_COrte_wrt x,y & u2,v2,u1,v1 are_COrte_wrt x,y
  proof
    assume
A3: u,v // u1,v1;
A4: now
      assume that
A5:   u=v and
A6:   u1=v1;
      take u2=0.V,v2=x;
A7:   Orte(x,y,u2),Orte(x,y,v2) // u,v by A5,ANALOAF:9;
      Orte(x,y,u2),Orte(x,y,v2) // u1,v1 by A6,ANALOAF:9;
      then
A8:   u2,v2,u1,v1 are_COrte_wrt x,y;
A9:   u2,v2,u,v are_COrte_wrt x,y by A7;
      u2<>v2 by A1,Lm4;
      hence thesis by A8,A9;
    end;
A10: now
      assume
A11:  u<>v;
      consider u2 such that
A12:  Orte(x,y,u2)=u by A1,Th15;
      consider v2 such that
A13:  Orte(x,y,v2)=v by A1,Th15;
      Orte(x,y,u2),Orte(x,y,v2) // u,v by A12,A13,ANALOAF:8;
      then
A14:  u2,v2,u,v are_COrte_wrt x,y;
      u2,v2,u1,v1 are_COrte_wrt x,y by A3,A12,A13;
      hence thesis by A11,A12,A13,A14;
    end;
    now
      assume
A15:  u1<>v1;
      consider u2 such that
A16:  Orte(x,y,u2)=u1 by A1,Th15;
      consider v2 such that
A17:  Orte(x,y,v2)=v1 by A1,Th15;
      Orte(x,y,u2),Orte(x,y,v2) // u1,v1 by A16,A17,ANALOAF:8;
      then
A18:  u2,v2,u1,v1 are_COrte_wrt x,y;
      Orte(x,y,u2),Orte(x,y,v2) // u,v by A3,A16,A17,ANALOAF:12;
      then u2,v2,u,v are_COrte_wrt x,y;
      hence thesis by A15,A16,A17,A18;
    end;
    hence thesis by A4,A10;
  end;
  (ex u2,v2 st u2<>v2 & u2,v2,u,v are_COrte_wrt x,y &
  u2,v2,u1,v1 are_COrte_wrt x,y) implies u,v // u1,v1
  by A1,Th13,ANALOAF:11;
  hence thesis by A2;
end;
