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 u,v,w ex u1 st w<>u1 & w,u1,u,v are_COrtm_wrt x,y
proof
  assume
A1: Gen x,y;
  let u,v,w;
A2: now
    assume
A3: u=v;
    take u1=w+x;
    Ortm(x,y,w),Ortm(x,y,u1) // u,v by A3,ANALOAF:9;
    then
A4: w,u1,u,v are_COrtm_wrt x,y;
    now
      assume w=u1;
      then x=0.V by RLVECT_1:9;
      hence contradiction by A1,Lm4;
    end;
    hence thesis by A4;
  end;
  now
    assume
A5: u<>v;
    consider u2 such that
A6: Ortm(x,y,u2)=u by A1,Th8;
    consider v2 such that
A7: Ortm(x,y,v2)=v by A1,Th8;
    take u1= (v2+w)-u2;
    u2,v2 // w,u1 by ANALOAF:16;
    then w,u1 // u2,v2 by ANALOAF:12;
    then Ortm(x,y,w),Ortm(x,y,u1) // Ortm(x,y,u2),Ortm(x,y,v2) by A1,Th17;
    then
A8: w,u1,u,v are_COrtm_wrt x,y by A6,A7;
    now
      assume w=u1;
      then w= w+(v2-u2) by RLVECT_1:def 3;
      then v2-u2=0.V by RLVECT_1:9;
      hence contradiction by A5,A6,A7,RLVECT_1:21;
    end;
    hence thesis by A8;
  end;
  hence thesis by A2;
end;
