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 for v,w,u1,v1,w1 holds not v,v1,w,u1 are_COrtm_wrt x,y &
  not v,v1,u1,w are_COrtm_wrt x,y & u1,w1,u1,w are_COrtm_wrt x,y
  implies ex u2 st
  (v,v1,v,u2 are_COrtm_wrt x,y or v,v1,u2,v are_COrtm_wrt x,y) &
  (u1,w1,u1,u2 are_COrtm_wrt x,y or u1,w1,u2,u1 are_COrtm_wrt x,y)
proof
  assume
A1: Gen x,y;
  let v,w,u1,v1,w1;
  consider u such that
A2: v<>u and
A3: v,v1,v,u are_COrtm_wrt x,y by A1,Th41;
  assume that
A4: not v,v1,w,u1 are_COrtm_wrt x,y and
A5: not v,v1,u1,w are_COrtm_wrt x,y and
A6: u1,w1,u1,w are_COrtm_wrt x,y;
A7: not Ortm(x,y,v),Ortm(x,y,v1) // w,u1 by A4;
A8: Ortm(x,y,v),Ortm(x,y,v1) // v,u by A3;
A9: Ortm(x,y,u1),Ortm(x,y,w1) // u1,w by A6;
A10: not Ortm(x,y,v),Ortm(x,y,v1) // u1,w by A5;
A11: v,u // Ortm(x,y,v),Ortm(x,y,v1) by A8,ANALOAF:12;
A12: u1,w // Ortm(x,y,u1),Ortm(x,y,w1) by A9,ANALOAF:12;
A13: u1<>w by A7,ANALOAF:9;
A14: not v,u // u1,w by A2,A10,A11,ANALOAF:11;
A15: not v,u // w,u1 by A2,A7,A11,ANALOAF:11;
  Gen x,y implies ex u,v st for w ex a,b st a*u + b*v=w
  proof
    assume
A16: Gen x,y;
    take x,y;
    thus thesis by A16,ANALMETR:def 1;
  end;
  then consider u2 such that
A17: v,u // v,u2 or v,u // u2,v and
A18: u1,w // u1,u2 or u1,w // u2,u1 by A1,A14,A15,ANALOAF:21;
  Ortm(x,y,v),Ortm(x,y,v1) // v,u2 or Ortm(x,y,v),Ortm(x,y,v1) // u2,v
  by A2,A11,A17,ANALOAF:11;
  then
A19: v,v1,v,u2 are_COrtm_wrt x,y or v,v1,u2,v are_COrtm_wrt x,y;
  Ortm(x,y,u1),Ortm(x,y,w1) // u1,u2 or Ortm(x,y,u1),Ortm(x,y,w1) // u2,u1
  by A12,A13,A18,ANALOAF:11;
  then u1,w1,u1,u2 are_COrtm_wrt x,y or u1,w1,u2,u1 are_COrtm_wrt x,y;
  hence thesis by A19;
end;
