reserve V for RealLinearSpace;
reserve p,q,u,v,w,y for VECTOR of V;
reserve a,b,c,d for Real;

theorem Th12:
  u,v // w,y implies v,u // y,w & w,y // u,v
proof
  assume
A1: u,v // w,y;
  now
    assume u<>v & w<>y;
    then consider a,b such that
A2: a*(v-u)=b*(y-w) and
A3: 0<a & 0<b by A1;
    a*(u-v)=-b*(y-w) by A2,Th3
      .=b*(w-y) by Th3;
    hence thesis by A2,A3;
  end;
  hence thesis;
end;
