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 Th4:
  are_Prop u,u1 & are_Prop v,v1 & are_Prop w,w1 & u,v,w are_LinDep
  implies u1,v1,w1 are_LinDep
proof
  assume that
A1: are_Prop u,u1 and
A2: are_Prop v,v1 and
A3: are_Prop w,w1 and
A4: u,v,w are_LinDep;
  consider b such that
A5: b<>0 and
A6: v = b*v1 by A2,Th1;
  consider a such that
A7: a<>0 and
A8: u = a*u1 by A1,Th1;
  consider d1,d2,d3 be Real such that
A9: d1*u + d2*v + d3*w = 0.V and
A10: d1<>0 or d2<>0 or d3<>0 by A4;
  consider c such that
A11: c <>0 and
A12: w = c*w1 by A3,Th1;
A13: d1*a<>0 or d2*b<>0 or d3*c <>0 by A7,A5,A11,A10,XCMPLX_1:6;
  0.V = (d1*a)*u1 + d2*(b*v1) + d3*(c*w1) by A8,A6,A12,A9,RLVECT_1:def 7
    .= (d1*a)*u1 + (d2*b)*v1 + d3*(c*w1) by RLVECT_1:def 7
    .= (d1*a)*u1 + (d2*b)*v1 + (d3*c)*w1 by RLVECT_1:def 7;
  hence thesis by A13;
end;
