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 & u,u1,v,v1 are_COrtm_wrt x,y & v,v1,w,w1 are_COrtm_wrt x,y
  & u,u1,u2,v2 are_COrtm_wrt x,y implies
  u2,v2,w,w1 are_COrtm_wrt x,y or v=v1 or u=u1
proof
  assume
A1: Gen x,y;
  assume that
A2: u,u1,v,v1 are_COrtm_wrt x,y and
A3: v,v1,w,w1 are_COrtm_wrt x,y and
A4: u,u1,u2,v2 are_COrtm_wrt x,y;
A5: Ortm(x,y,u),Ortm(x,y,u1) // v,v1 by A2;
A6: Ortm(x,y,v),Ortm(x,y,v1) // w,w1 by A3;
A7: Ortm(x,y,u),Ortm(x,y,u1) // u2,v2 by A4;
  now
    assume that
A8: u<>u1 and
A9: v<>v1;
    v,v1 // u2,v2 by A1,A5,A7,A8,Th6,ANALOAF:11;
    then Ortm(x,y,v),Ortm(x,y,v1) // Ortm(x,y,u2),Ortm(x,y,v2) by A1,Th17;
    then Ortm(x,y,u2),Ortm(x,y,v2) // w,w1 by A1,A6,A9,Th6,ANALOAF:11;
    hence thesis;
  end;
  hence thesis;
end;
