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

theorem Th10:
  u,v // v,u implies u=v
proof
  assume
A1: u,v // v,u;
  assume
A2: u<>v;
  then consider a,b such that
A3: a*(v-u)=b*(u-v) and
A4: 0<a & 0<b by A1;
  a*(v-u)=-b*(v-u) by A3,Th3;
  then b*(v-u)+a*(v-u)=0.V by RLVECT_1:5;
  then (b+a)*(v-u)=0.V by RLVECT_1:def 6;
  then v-u=0.V or b+a=0 by RLVECT_1:11;
  then 0.V=(-u)+v by A4;
  then v=-(-u) by RLVECT_1:def 10
    .=u by RLVECT_1:17;
  hence contradiction by A2;
end;
