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 Th6:
  o,u,u2 are_LinDep & not are_Prop o,u & not are_Prop o,u2 & not
are_Prop u,u2 & o,u,u2 are_Prop_Vect implies (ex a1,b1 st b1*u2=o+a1*u & a1<>0
  & b1<>0) & ex a2,c2 st u2=c2*o+a2*u & c2<>0 & a2<>0
proof
  assume that
A1: o,u,u2 are_LinDep and
A2: not are_Prop o,u and
A3: not are_Prop o,u2 and
A4: not are_Prop u,u2 and
A5: o,u,u2 are_Prop_Vect;
  consider a,b,c such that
A6: a*o + b*u + c*u2 = 0.V and
A7: a<>0 or b<>0 or c <>0 by A1;
  u is not zero by A5;
  then
A8: u <> 0.V;
  u2 is not zero by A5;
  then
A9: u2 <>0.V;
  o is not zero by A5;
  then
A10: o <> 0.V;
A11: a<>0 & b<>0 & c <>0
  proof
A12: now
      assume
A13:  b = 0;
      then 0.V = a*o + 0.V + c*u2 by A6,RLVECT_1:10
        .= a*o + c*u2 by RLVECT_1:4;
      then a*o = -c*u2 by RLVECT_1:6
        .= c*(-u2) by RLVECT_1:25;
      then
A14:  a*o = (-c)*u2 by RLVECT_1:24;
A15:  a<>0 & c <>0
      proof
A16:    now
          assume
A17:      c = 0;
          then 0.V = a*o + 0*u + 0.V by A6,A13,RLVECT_1:10
            .= a*o + 0*u by RLVECT_1:4
            .= a*o + 0.V by RLVECT_1:10
            .= a*o by RLVECT_1:4;
          hence contradiction by A7,A10,A13,A17,RLVECT_1:11;
        end;
A18:    now
          assume
A19:      a = 0;
          then 0.V = 0.V + 0*u + c*u2 by A6,A13,RLVECT_1:10
            .= 0*u + c*u2 by RLVECT_1:4
            .= 0.V + c*u2 by RLVECT_1:10
            .= c*u2 by RLVECT_1:4;
          hence contradiction by A7,A9,A13,A19,RLVECT_1:11;
        end;
        assume not thesis;
        hence contradiction by A18,A16;
      end;
      then -c <>0;
      hence contradiction by A3,A15,A14;
    end;
A20: now
      assume
A21:  a = 0;
      then 0.V = 0.V + b*u + c*u2 by A6,RLVECT_1:10
        .= b*u + c*u2 by RLVECT_1:4;
      then b*u = -c*u2 by RLVECT_1:6
        .= c*(-u2) by RLVECT_1:25;
      then
A22:  b*u = (-c)*u2 by RLVECT_1:24;
A23:  b<>0 & c <>0
      proof
A24:    now
          assume
A25:      c = 0;
          then 0.V = 0.V + b*u + 0*u2 by A6,A21,RLVECT_1:10
            .= b*u + 0*u2 by RLVECT_1:4
            .= b*u + 0.V by RLVECT_1:10
            .= b*u by RLVECT_1:4;
          hence contradiction by A7,A8,A21,A25,RLVECT_1:11;
        end;
A26:    now
          assume
A27:      b = 0;
          then 0.V = 0.V + 0*u + c*u2 by A6,A21,RLVECT_1:10
            .= 0*u + c*u2 by RLVECT_1:4
            .= 0.V + c*u2 by RLVECT_1:10
            .= c*u2 by RLVECT_1:4;
          hence contradiction by A7,A9,A21,A27,RLVECT_1:11;
        end;
        assume not thesis;
        hence contradiction by A26,A24;
      end;
      then -c <>0;
      hence contradiction by A4,A23,A22;
    end;
A28: now
      assume
A29:  c = 0;
      then 0.V = a*o + b*u + 0.V by A6,RLVECT_1:10
        .= a*o + b*u by RLVECT_1:4;
      then a*o = -b*u by RLVECT_1:6
        .= b*(-u) by RLVECT_1:25;
      then
A30:  a*o = (-b)*u by RLVECT_1:24;
A31:  a<>0 & b<>0
      proof
A32:    now
          assume
A33:      b = 0;
          then 0.V = a*o + 0*u + 0.V by A6,A29,RLVECT_1:10
            .= a*o + 0*u by RLVECT_1:4
            .= a*o + 0.V by RLVECT_1:10
            .= a*o by RLVECT_1:4;
          hence contradiction by A7,A10,A29,A33,RLVECT_1:11;
        end;
A34:    now
          assume
A35:      a = 0;
          then 0.V = 0.V + b*u + 0*u2 by A6,A29,RLVECT_1:10
            .= b*u + 0*u2 by RLVECT_1:4
            .= b*u + 0.V by RLVECT_1:10
            .= b*u by RLVECT_1:4;
          hence contradiction by A7,A8,A29,A35,RLVECT_1:11;
        end;
        assume not thesis;
        hence contradiction by A34,A32;
      end;
      then -b<>0;
      hence contradiction by A2,A31,A30;
    end;
    assume not thesis;
    hence contradiction by A20,A12,A28;
  end;
  then
A36: c" <> 0 by XCMPLX_1:202;
  a" <> 0 by A11,XCMPLX_1:202;
  then
A37: a"*b <> 0 & -(a"*c) <> 0 by A11,XCMPLX_1:6;
  a"*(-c*u2) = a"*(a*o + b*u) by A6,RLVECT_1:6
    .= a"*(a*o) + a"*(b*u) by RLVECT_1:def 5
    .= (a"*a)*o + a"*(b*u) by RLVECT_1:def 7
    .= (a"*a)*o + (a"*b)*u by RLVECT_1:def 7
    .= 1*o + (a"*b)*u by A11,XCMPLX_0:def 7
    .= o + (a"*b)*u by RLVECT_1:def 8;
  then o + (a"*b)*u = a"*(c*(-u2)) by RLVECT_1:25
    .= (a"*c)*(-u2) by RLVECT_1:def 7
    .= (-(a"*c))*u2 by RLVECT_1:24;
  hence ex a1,b1 st b1*u2=o+a1*u & a1<>0 & b1<>0 by A37;
  -b <> 0 by A11;
  then
A38: (c"*(-b)) <> 0 by A36,XCMPLX_1:6;
  c*u2 = -(a*o + b*u) by A6,RLVECT_1:def 10
    .= -(a*o) + (-(b*u)) by RLVECT_1:31
    .= (-a)*o + (-(b*u)) by Lm8
    .= (-a)*o + (-b)*u by Lm8;
  then c"*(c*u2) = c"*((-a)*o) + c"*((-b)*u) by RLVECT_1:def 5
    .= (c"*(-a))*o + c"*((-b)*u) by RLVECT_1:def 7
    .= (c"*(-a))*o + (c"*(-b))*u by RLVECT_1:def 7;
  then
A39: (c"*(-a))*o + (c"*(-b))*u = (c"*c)*u2 by RLVECT_1:def 7
    .= 1*u2 by A11,XCMPLX_0:def 7
    .= u2 by RLVECT_1:def 8;
  -a <> 0 by A11;
  then (c"*(-a)) <> 0 by A36,XCMPLX_1:6;
  hence thesis by A39,A38;
end;
