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

theorem Th17:
  (ex p,q st p<>q) implies for u,v,w ex y st u,v // w,y & u,w // v ,y & v<>y
proof
  given p,q such that
A1: p<>q;
  let u,v,w;
A2: now
    assume
A3: u<>w;
    take y=(v+w)-u;
A4: now
      assume v=y;
      then v=v+(w-u) by RLVECT_1:def 3;
      then w-u=0.V by RLVECT_1:9;
      hence contradiction by A3,RLVECT_1:21;
    end;
    u,v // w,y & u,w // v,y by Th16;
    hence thesis by A4;
  end;
  now
    assume
A5: u=w;
A6: now
      assume u=v;
      then
A7:   u,v // w,p & u,v // w,q;
A8:   v<>p or v<>q by A1;
      u,w // v,p & u,w // v,q by A5;
      hence thesis by A8,A7;
    end;
    u,v // w,u & u,w // v,u by A5;
    hence thesis by A6;
  end;
  hence thesis by A2;
end;
