reserve V for RealLinearSpace;
reserve u,u1,u2,v,v1,v2,w,w1,y for VECTOR of V;
reserve a,a1,a2,b,b1,b2,c1,c2 for Real;
reserve x,z for set;

theorem Th14:
  u,v // u1,v1 implies u,v // (u#u1),(v#v1)
proof
  assume
A1: u,v // u1,v1;
  per cases;
  suppose
    u=v or u1 = v1;
    hence thesis by Th8,ANALOAF:9;
  end;
  suppose
A2: u<>v & u1<>v1;
    set p=u#u1,q=v#v1;
    consider a,b such that
A3: 0<a & 0<b and
A4: a*(v-u) = b*(v1-u1) by A1,A2,ANALOAF:def 1;
A5: 0<a+b & 0<b*2 by A3,XREAL_1:34,129;
    (b*2)*(q-p) = b*(2*(q-p)) by RLVECT_1:def 7
      .= b*((1+1)*q - 2*p) by RLVECT_1:34
      .= b*((1*q+1*q) - 2*p) by RLVECT_1:def 6
      .= b*((q+1*q) - 2*p) by RLVECT_1:def 8
      .= b*((q+q) - 2*p) by RLVECT_1:def 8
      .= b*((v+v1) - 2*p) by Def2
      .= b*(v+(v1 - (1+1)*p)) by RLVECT_1:def 3
      .= b*(v+(v1 - (1*p+1*p))) by RLVECT_1:def 6
      .= b*(v+(v1 - (p+1*p))) by RLVECT_1:def 8
      .= b*(v+(v1 - (p+p))) by RLVECT_1:def 8
      .= b*(v+(v1 - (u+u1))) by Def2
      .= b*(v+((v1 - u1)-u)) by RLVECT_1:27
      .= b*((v+(v1 - u1))-u) by RLVECT_1:def 3
      .= b*((v1 - u1)+(v-u)) by RLVECT_1:def 3
      .= a*(v - u)+b*(v-u) by A4,RLVECT_1:def 5
      .= (a+b)*(v-u) by RLVECT_1:def 6;
    hence thesis by A5,ANALOAF:def 1;
  end;
end;
