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 Th4:
  (for w ex a,b,c,d st w = a*p + b*q + c*r + d*s) & (for a,b,c,d st
a*p + b*q + c*r + d*s = 0.V holds a = 0 & b = 0 & c = 0 & d = 0) implies for u,
  v st u is not zero & v is not zero ex y,w st u,v,w are_LinDep & q,r,y
  are_LinDep & p,w,y are_LinDep & y is not zero & w is not zero
proof
  assume that
A1: for w ex a,b,c,d st w = a*p + b*q + c*r + d*s and
A2: for a,b,c,d st a*p + b*q + c*r + d*s = 0.V holds a = 0 & b = 0 & c =
  0 & d = 0;
A3: p is not zero by A2,Th2;
  let u,v such that
A4: u is not zero and
A5: v is not zero;
  consider a1,b1,c1,d1 such that
A6: u = a1*p + b1*q + c1*r + d1*s by A1;
  not p,q,r are_LinDep by A2,Th2;
  then
A7: not are_Prop q,r by ANPROJ_1:11;
A8: q is not zero by A2,Th2;
  consider a2,b2,c2,d2 such that
A9: v = a2*p + b2*q + c2*r + d2*s by A1;
A10: a3*u + b3*v = (a3*a1 + b3*a2)*p + (a3*b1 + b3*b2)*q + (a3*c1 + b3*c2)*r
  + (a3*d1 + b3*d2)*s
  proof
    a3*u = (a3*a1)*p + (a3*b1)*q + (a3*c1)*r + (a3*d1)*s by A6,Lm4;
    hence
    a3*u + b3*v = ((a3*a1)*p + (a3*b1)*q + (a3*c1)*r + (a3*d1)*s) + ((b3*
    a2)*p + (b3*b2)*q + (b3*c2)*r + (b3*d2)*s) by A9,Lm4
      .= ((a3*a1)*p + (b3*a2)*p) + ((a3*b1)*q + (b3*b2)*q) + ((a3*c1)*r + (
    b3*c2)*r) + ((a3*d1)*s + (b3*d2)*s) by Lm5
      .= (a3*a1+b3*a2)*p + ((a3*b1)*q + (b3*b2)*q) + ((a3*c1)*r + (b3*c2)*r)
    + ((a3*d1)*s + (b3*d2)*s) by RLVECT_1:def 6
      .= (a3*a1+b3*a2)*p + (a3*b1+b3*b2)*q + ((a3*c1)*r + (b3*c2)*r) + ((a3*
    d1)*s + (b3*d2)*s) by RLVECT_1:def 6
      .= (a3*a1+b3*a2)*p + (a3*b1+b3*b2)*q + (a3*c1+ b3*c2)*r + ((a3*d1)*s +
    (b3*d2)*s) by RLVECT_1:def 6
      .= (a3*a1+b3*a2)*p + (a3*b1+b3*b2)*q + (a3*c1+b3*c2)*r + (a3*d1+b3*d2)
    *s by RLVECT_1:def 6;
  end;
A11: r is not zero by A2,Th2;
A12: now
    assume
A13: not are_Prop u,v;
    ex w st (w is not zero & u,v,w are_LinDep & ex A,B,C being Real
      st w = A*p + B*q + C*r)
    proof
A14:  now
        set a3 = -d2*d1",b3=1, w = a3*u+b3*v;
        assume that
A15:    d1<>0 and
A16:    d2<>0;
        set A = a3*a1 + b3*a2, B = a3*b1 + b3*b2, C = a3*c1 + b3*c2;
A17:    A<>0 or B<>0 or C<>0
        proof
A18:      d2*d1" <> 0
          proof
            assume not thesis;
            then d1"=0 by A16,XCMPLX_1:6;
            hence contradiction by A15,XCMPLX_1:202;
          end;
A19:      d2*d1"*d1 = d2*(d1"*d1) .= d2*1 by A15,XCMPLX_0:def 7
            .= d2;
          assume
A20:      not thesis;
          then
A21:      --d2*d1"*c1 = c2;
          --d2*d1"*a1 = a2 & --d2*d1"*b1 = b2 by A20;
          then (d2*d1")*u = v by A6,A9,A21,A19,Lm4;
          hence contradiction by A13,A18,ANPROJ_1:1;
        end;
        a3*d1 + b3*d2 = -d2*(d1"*d1) + d2 .= -d2*1 + d2 by A15,XCMPLX_0:def 7
          .= 0;
        then
A22:    w = (a3*a1 + b3*a2)*p + (a3*b1 + b3*b2)*q + (a3*c1 + b3*c2 )*r +
        0*s by A10
          .= (a3*a1 + b3*a2)*p + (a3*b1 + b3*b2)*q + (a3*c1 + b3*c2)*r + 0.V
        by RLVECT_1:10
          .= (a3*a1 + b3*a2)*p + (a3*b1 + b3*b2)*q + (a3*c1 + b3*c2)*r by
RLVECT_1:4;
        then
A23:    w = A*p+B*q+C*r+0.V by RLVECT_1:4
          .= A*p+B*q+C*r+0*s by RLVECT_1:10;
A24:    w is not zero
        by A2,A23,A17;
        u,v,w are_LinDep by A4,A5,A13,ANPROJ_1:6;
        hence thesis by A22,A24;
      end;
A25:  now
        assume
A26:    d2=0;
        take w = v;
A27:    u,v,w are_LinDep by ANPROJ_1:11;
        w = a2*p + b2*q + c2*r + 0.V by A9,A26,RLVECT_1:10
          .= a2*p + b2*q + c2*r by RLVECT_1:4;
        hence thesis by A5,A27;
      end;
      now
        assume
A28:    d1=0;
        take w = u;
A29:    u,v,w are_LinDep by ANPROJ_1:11;
        w = a1*p + b1*q + c1*r + 0.V by A6,A28,RLVECT_1:10
          .= a1*p + b1*q + c1*r by RLVECT_1:4;
        hence thesis by A4,A29;
      end;
      hence thesis by A25,A14;
    end;
    then consider w such that
A30: w is not zero and
A31: u,v,w are_LinDep and
A32: ex A,B,C being Real st w = A*p + B*q + C*r;
    consider A,B,C being Real such that
A33: w = A*p + B*q + C*r by A32;
A34: now
      set b = 1, a = -A;
      set y = a*p + b*w;
A35:  y = (a+b*A)*p + (b*B)*q + (b*C)*r by A33,Lm7
        .= 0.V + (1*B)*q + (1*C)*r by RLVECT_1:10
        .= B*q + C*r by RLVECT_1:4;
      assume
A36:  not are_Prop p,w;
      then
A37:  p,w,y are_LinDep by A3,A30,ANPROJ_1:6;
A38:  B<>0 or C<>0
      proof
        assume not thesis;
        then
A39:    w = A*p + 0.V + 0*r by A33,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;
        A<>0
        by A39,RLVECT_1:10,A30;
        hence contradiction by A36,A39,ANPROJ_1:1;
      end;
A40:  y is not zero
      proof
        assume not thesis;
        then 0.V = B*q + C*r by A35
          .= 0.V + B*q + C*r by RLVECT_1:4
          .= 0*p + B*q + C*r by RLVECT_1:10
          .= 0*p + B*q + C*r + 0.V by RLVECT_1:4
          .= 0*p + B*q + C*r + 0*s by RLVECT_1:10;
        hence contradiction by A2,A38;
      end;
      q,r,y are_LinDep by A8,A11,A7,A35,ANPROJ_1:6;
      hence thesis by A30,A31,A40,A37;
    end;
    now
      assume are_Prop p,w;
      then q,r,q are_LinDep & p,w,q are_LinDep by ANPROJ_1:11;
      hence thesis by A8,A30,A31;
    end;
    hence thesis by A34;
  end;
  now
    assume are_Prop u,v;
    then
A41: u,v,p are_LinDep by ANPROJ_1:11;
    q,r,q are_LinDep & p,p,q are_LinDep by ANPROJ_1:11;
    hence thesis by A3,A8,A41;
  end;
  hence thesis by A12;
end;
