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 or u,v // v1,u1) iff ex a,b st a*(v-u) = b*(v1-u1)
  & (a<>0 or b<>0)
proof
A1: now
    let w,y,w1,y1 be VECTOR of V;
    given a,b such that
A2: a*(y-w) = b*(y1-w1) & a=0 and
A3: b<>0;
    0.V = b*(y1-w1) by A2,RLVECT_1:10;
    then y1-w1 = 0.V by A3,RLVECT_1:11;
    then y1 = w1 by RLVECT_1:21;
    hence w,y // w1,y1 by ANALOAF:9;
  end;
A4: now
    let w,y,w1,y1 be VECTOR of V;
    given a,b such that
A5: a*(y-w) = b*(y1-w1) and
A6: 0<a and
A7: b<0;
A8: a*(y-w) = b*(-(w1-y1)) by A5,RLVECT_1:33
      .= (-b)*(w1-y1) by RLVECT_1:24;
    0<-b by A7,XREAL_1:58;
    hence w,y // y1,w1 by A6,A8,ANALOAF:def 1;
  end;
A9: now
    given a,b such that
A10: a*(v-u) = b*(v1-u1) and
A11: a<>0 or b<>0;
A12: now
A13:  now
        assume a<0 & b<0;
        then
A14:    0< -a & 0< -b by XREAL_1:58;
        (-a)*(u-v) = a*(-(u-v)) by RLVECT_1:24
          .= b*(v1-u1) by A10,RLVECT_1:33
          .= b*(-(u1-v1)) by RLVECT_1:33
          .= (-b)*(u1-v1) by RLVECT_1:24;
        then v,u // v1,u1 by A14,ANALOAF:def 1;
        hence u,v // u1,v1 or u,v // v1,u1 by ANALOAF:12;
      end;
A15:  now
        assume a<0 & 0<b;
        then u1,v1 // v,u by A4,A10;
        then v,u // u1, v1 by ANALOAF:12;
        hence u,v // u1,v1 or u,v // v1,u1 by ANALOAF:12;
      end;
      assume
A16:  a<>0 & b<>0;
      0<a & b<0 implies ( u,v // u1,v1 or u,v // v1,u1) by A4,A10;
      hence u,v // u1,v1 or u,v // v1,u1 by A10,A16,A15,A13,ANALOAF:def 1;
    end;
    now
      assume b=0;
      then u1,v1 // u,v by A1,A10,A11;
      hence u,v // u1,v1 or u,v // v1,u1 by ANALOAF:12;
    end;
    hence u,v // u1,v1 or u,v // v1,u1 by A1,A10,A12;
  end;
A17: now
    let w,y,w1,y1 be VECTOR of V such that
A18: w,y // w1,y1;
A19: now
      assume w=y;
      then 1*(y-w) = 0.V by RLVECT_1:10,15
        .= 0*(y1-w1) by RLVECT_1:10;
      hence ex a,b st a*(y-w) = b*(y1-w1) & (a<>0 or b<>0);
    end;
A20: now
      assume w1=y1;
      then 1*(y1-w1) = 0.V by RLVECT_1:10,15
        .= 0*(y-w) by RLVECT_1:10;
      hence ex a,b st a*(y-w) = b*(y1-w1) & (a<>0 or b<>0);
    end;
    (ex a,b st 0<a & 0<b & a*(y-w)=b*(y1-w1)) implies ex a,b st a*(y-w) =
    b*(y1-w1) & (a<>0 or b<>0);
    hence ex a,b st a*(y-w) = b*(y1-w1) & (a<>0 or b<>0) by A18,A19,A20,
ANALOAF:def 1;
  end;
  now
    assume u,v // v1,u1;
    then consider a,b such that
A21: a*(v-u) = b*(u1-v1) and
A22: a<>0 or b<>0 by A17;
A23: a<>0 or -b<>0 by A22;
    (-b)*(v1-u1) = b*(-(v1-u1)) by RLVECT_1:24
      .= a*(v-u) by A21,RLVECT_1:33;
    hence ex a,b st a*(v-u) = b*(v1-u1) & (a<>0 or b<>0 ) by A23;
  end;
  hence thesis by A17,A9;
end;
