reserve V for RealLinearSpace;
reserve p,q,r,u,v,w,y,u1,v1,w1 for Element of V;
reserve a,b,c,d,a1,b1,c1,a2,b2,c2,a3,b3,e,f for Real;

theorem Th1:
  are_Prop p,q iff ex a st a<>0 & p = a*q
proof
A1: now
    assume are_Prop p,q;
    then consider a,b such that
A2: a*p = b*q and
A3: a<>0 and
A4: b<>0;
A5: a" <> 0 by A3,XCMPLX_1:202;
    p = 1*p by RLVECT_1:def 8
      .= (a"*a)*p by A3,XCMPLX_0:def 7
      .= (a")*(b*q) by A2,RLVECT_1:def 7
      .= (a"*b)*q by RLVECT_1:def 7;
    hence ex a st a<>0 & p = a*q by A4,A5,XCMPLX_1:6;
  end;
  now
    given a such that
A6: a<>0 and
A7: p = a*q;
    1*p = a*q by A7,RLVECT_1:def 8;
    hence are_Prop p,q by A6;
  end;
  hence thesis by A1;
end;
