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 Th16:
  Gen x,y implies (u,v // u1,v1 implies
  Orte(x,y,u),Orte(x,y,v) // Orte(x,y,u1),Orte(x,y,v1))
proof
  assume
A1: Gen x,y;
  assume
A2: u,v // u1,v1;
  now
    assume that
A3: u<>v and
A4: u1<>v1;
    consider a,b such that
A5: 0<a and
A6: 0<b and
A7: a*(v-u)=b*(v1-u1) by A2,A3,A4,ANALOAF:def 1;
    a*(Orte(x,y,v)-Orte(x,y,u)) = a*(Orte(x,y,v-u)) by A1,Th11
      .= Orte(x,y,b*(v1-u1)) by A1,A7,Th12
      .= b*(Orte(x,y,v1-u1)) by A1,Th12
      .=b*(Orte(x,y,v1)-Orte(x,y,u1)) by A1,Th11;
    hence thesis by A5,A6,ANALOAF:def 1;
  end;
  hence thesis by ANALOAF:9;
end;
