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
  u,v,q are_LinDep & w,y,q are_LinDep & u,w,p are_LinDep & v,y,p
are_LinDep & u,y,r are_LinDep & v,w,r are_LinDep & p,q,r are_LinDep & p is not
  zero & q is not zero & r is not zero implies (u,v,y are_LinDep or u,v,w
  are_LinDep or u,w,y are_LinDep or v,w,y are_LinDep)
proof
  assume that
A1: u,v,q are_LinDep and
A2: w,y,q are_LinDep and
A3: u,w,p are_LinDep and
A4: v,y,p are_LinDep and
A5: u,y,r are_LinDep and
A6: v,w,r are_LinDep and
A7: p,q,r are_LinDep and
A8: p is not zero and
A9: q is not zero and
A10: r is not zero;
  assume
A11: not thesis;
  then
A12: v is not zero by Th12;
A13: w is not zero by A11,Th12;
A14: y is not zero by A11,Th12;
A15: u is not zero by A11,Th12;
  not are_Prop v,y by A11,Th12;
  then consider a19,b19 being Real such that
A16: p = a19*v + b19*y by A4,A12,A14,Th6;
  not are_Prop u,v by A11,Th12;
  then consider a2,b2 such that
A17: q = a2*u + b2*v by A1,A15,A12,Th6;
  not are_Prop v,w by A11,Th12;
  then consider a39,b39 being Real such that
A18: r = a39*v + b39*w by A6,A12,A13,Th6;
  not are_Prop u,w by A11,Th12;
  then consider a1,b1 such that
A19: p = a1*u + b1*w by A3,A15,A13,Th6;
  not are_Prop w,y by A11,Th12;
  then consider a29,b29 being Real such that
A20: q = a29*w + b29*y by A2,A13,A14,Th6;
  not are_Prop y,u by A11,Th12;
  then consider a3,b3 such that
A21: r = a3*u + b3*y by A5,A15,A14,Th6;
  consider A,B,C being Real such that
A22: A*p + B*q + C*r = 0.V and
A23: A<>0 or B<>0 or C<>0 by A7;
A24: 0.V = (A*a1 + C*a3)*u + (A*b1 + B*a29)*w + (B*b29 + C*b3)*y by A19,A20,A21
,A22,Lm9;
  then
A25: A*a1 + C*a3 = 0 by A11;
A26: 0.V = C*(a39*v + b39*w) + B*(a29*w + b29*y) + A*(a19*v + b19*y) by A16,A20
,A18,A22,RLVECT_1:def 3
    .= (C*a39 + A*a19)*v + (C*b39 + B*a29)*w + (B*b29 + A*b19)*y by Lm9;
  then
A27: C*a39 + A*a19 = 0 by A11;
A28: 0.V = (B*a2 + C*a3)*u + (B*b2 + A*a19)*v + (A*b19 + C*b3)*y by A16,A17,A21
,A22,Lm9;
  then
A29: B*a2 + C*a3 = 0 by A11;
A30: 0.V = (B*a2 + A*a1)*u + (B*b2 + C*a39)*v + (C*b39 + A*b1)*w by A19,A17,A18
,A22,Lm9;
  then
A31: B*a2 + A*a1 = 0 by A11;
A32: C*b39 + B*a29 = 0 by A11,A26;
A33: C*b39 + A*b1 = 0 by A11,A30;
A34: B*b29 + A*b19 = 0 by A11,A26;
A35: A*b19 + C*b3 = 0 by A11,A28;
A36: B*b29 + C*b3 = 0 by A11,A24;
A37: now
    assume
A38: C<>0;
    then a3 = 0 by A25,A29,A31,XCMPLX_1:6;
    then r = 0*u + 0*y by A21,A36,A35,A34,A38,XCMPLX_1:6
      .= 0.V + 0*y by RLVECT_1:10
      .= 0.V + 0.V by RLVECT_1:10
      .= 0.V;
    hence contradiction by A10;
  end;
A39: B*b2 + C*a39 = 0 by A11,A30;
A40: B*b2 + A*a19 = 0 by A11,A28;
A41: now
    assume
A42: B<>0;
    then a2 = 0 by A25,A29,A31,XCMPLX_1:6;
    then q = 0*u + 0*v by A17,A40,A39,A27,A42,XCMPLX_1:6
      .= 0.V + 0*v by RLVECT_1:10
      .= 0.V + 0.V by RLVECT_1:10
      .= 0.V;
    hence contradiction by A9;
  end;
A43: A*b1 + B*a29= 0 by A11,A24;
  now
    assume
A44: A<>0;
    then a1 = 0 by A25,A29,A31,XCMPLX_1:6;
    then p = 0*u + 0*w by A19,A43,A33,A32,A44,XCMPLX_1:6
      .= 0.V + 0*w by RLVECT_1:10
      .= 0.V + 0.V by RLVECT_1:10
      .= 0.V;
    hence contradiction by A8;
  end;
  hence thesis by A23,A41,A37;
end;
