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 Th19:
  Gen x,y implies (u,u1,v,v1 are_COrtm_wrt x,y implies
  v,v1,u,u1 are_COrtm_wrt x,y)
proof
  assume
A1: Gen x,y;
  assume u,u1,v,v1 are_COrtm_wrt x,y;
  then Ortm(x,y,u),Ortm(x,y,u1) // v,v1;
  then v,v1 // Ortm(x,y,u),Ortm(x,y,u1) by ANALOAF:12;
  then Ortm(x,y,v),Ortm(x,y,v1) //
  Ortm(x,y,Ortm(x,y,u)),Ortm(x,y,Ortm(x,y,u1)) by A1,Th17;
  then Ortm(x,y,v),Ortm(x,y,v1) // u,Ortm(x,y,Ortm(x,y,u1)) by A1,Th7;
  then Ortm(x,y,v),Ortm(x,y,v1) // u,u1 by A1,Th7;
  hence thesis;
end;
