reserve V for RealLinearSpace,
  o,p,q,r,s,u,v,w,y,y1,u1,v1,w1,u2,v2,w2 for Element of V,
  a,b,c,d,a1,b1,c1,d1,a2,b2,c2,d2,a3,b3,c3,d3 for Real,
  z for set;

theorem Th7:
  p,q,r are_LinDep & not are_Prop p,q & p,q,r are_Prop_Vect implies
  ex a,b st r = a*p + b*q
proof
  assume that
A1: p,q,r are_LinDep and
A2: not are_Prop p,q and
A3: p,q,r are_Prop_Vect;
  consider a,b,c such that
A4: a*p + b*q + c*r = 0.V and
A5: a<>0 or b<>0 or c <>0 by A1;
  q is not zero by A3;
  then
A6: q <> 0.V;
  p is not zero by A3;
  then
A7: p <> 0.V;
A8: c <>0
  proof
    assume
A9: not thesis;
    then 0.V = a*p + b*q + 0.V by A4,RLVECT_1:10
      .= a*p + b*q by RLVECT_1:4;
    then
A10: a*p = -(b*q) by RLVECT_1:6
      .= (-b)*q by Lm8;
A11: a<>0 & b<>0
    proof
A12:  now
        assume
A13:    b = 0;
        then 0.V =a*p + 0.V + 0*r by A4,A9,RLVECT_1:10
          .= a*p + 0.V + 0.V by RLVECT_1:10
          .= a*p + 0.V by RLVECT_1:4
          .= a*p by RLVECT_1:4;
        hence contradiction by A7,A5,A9,A13,RLVECT_1:11;
      end;
A14:  now
        assume
A15:    a = 0;
        then 0.V = 0.V + b*q + 0*r by A4,A9,RLVECT_1:10
          .= 0.V + b*q + 0.V by RLVECT_1:10
          .= b*q + 0.V by RLVECT_1:4
          .= b*q by RLVECT_1:4;
        hence contradiction by A6,A5,A9,A15,RLVECT_1:11;
      end;
      assume not thesis;
      hence contradiction by A14,A12;
    end;
    then -b <> 0;
    hence contradiction by A2,A11,A10;
  end;
  c*r = -(a*p + b*q) by A4,RLVECT_1:def 10
    .= -(a*p) + (-(b*q)) by RLVECT_1:31
    .= (-a)*p + (-(b*q)) by Lm8
    .= (-a)*p + (-b)*q by Lm8;
  then c"*(c*r) = c"*((-a)*p) + c"*((-b)*q) by RLVECT_1:def 5
    .= (c"*(-a))*p + c"*((-b)*q) by RLVECT_1:def 7
    .= (c"*(-a))*p + (c"*(-b))*q by RLVECT_1:def 7;
  then (c"*(-a))*p + (c"*(-b))*q = (c"*c)*r by RLVECT_1:def 7
    .= 1*r by A8,XCMPLX_0:def 7
    .= r by RLVECT_1:def 8;
  hence thesis;
end;
