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 Th17:
  Gen x,y implies (u,v // u1,v1 implies
  Ortm(x,y,u),Ortm(x,y,v) // Ortm(x,y,u1),Ortm(x,y,v1))
proof
  assume
A1: Gen x,y;
  assume
A2: u,v // u1,v1;
  now
    assume
A3: u<>v;
    now
      assume u1<>v1;
      then consider a,b such that
A4:   0<a and
A5:   0<b and
A6:   a*(v-u)=b*(v1-u1) by A2,A3,ANALOAF:def 1;
      a*(Ortm(x,y,v)-Ortm(x,y,u)) = a*(Ortm(x,y,v-u)) by A1,Th5
        .= Ortm(x,y,b*(v1-u1)) by A1,A6,Th2
        .= b*(Ortm(x,y,v1-u1)) by A1,Th2
        .=b*(Ortm(x,y,v1)-Ortm(x,y,u1)) by A1,Th5;
      hence thesis by A4,A5,ANALOAF:def 1;
    end;
    hence thesis by ANALOAF:9;
  end;
  hence thesis by ANALOAF:9;
end;
