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 Th2:
  are_Prop p,u & are_Prop u,q implies are_Prop p,q
proof
  assume that
A1: are_Prop p,u and
A2: are_Prop u,q;
  consider a,b such that
A3: a*p = b*u and
A4: a<>0 and
A5: b<>0 by A1;
  consider c,d such that
A6: c*u = d*q and
A7: c <>0 and
A8: d<>0 by A2;
  b" <>0 by A5,XCMPLX_1:202;
  then b"*a<>0 by A4,XCMPLX_1:6;
  then
A9: c*(b"*a)<>0 by A7,XCMPLX_1:6;
  (b"*a)*p = (b")*(b*u) by A3,RLVECT_1:def 7
    .= (b"*b)*u by RLVECT_1:def 7
    .= 1*u by A5,XCMPLX_0:def 7
    .= u by RLVECT_1:def 8;
  then d*q = (c*(b"*a))*p by A6,RLVECT_1:def 7;
  hence thesis by A8,A9;
end;
