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
  not u,v,w are_LinDep & u,v,p are_LinDep & v,w,q are_LinDep implies ex
  y st u,w,y are_LinDep & p,q,y are_LinDep & y is not zero
proof
  assume that
A1: not u,v,w are_LinDep and
A2: u,v,p are_LinDep and
A3: v,w,q are_LinDep;
A4: v is not zero by A1,Th12;
A5: w is not zero by A1,Th12;
A6: now
A7: now
      assume not q is not zero;
      then q = 0.V;
      then
A8:   p,q,w are_LinDep by Th10;
      u,w,w are_LinDep by Th11;
      hence thesis by A5,A8;
    end;
A9: now
      assume not p is not zero;
      then p = 0.V;
      then
A10:  p,q,w are_LinDep by Th10;
      u,w,w are_LinDep by Th11;
      hence thesis by A5,A10;
    end;
A11: now
      assume are_Prop p,q;
      then
A12:  p,q,w are_LinDep by Th11;
      u,w,w are_LinDep by Th11;
      hence thesis by A5,A12;
    end;
    assume are_Prop p,q or not p is not zero or not q is not zero;
    hence thesis by A11,A9,A7;
  end;
A13: u is not zero by A1,Th12;
  not are_Prop u,v by A1,Th12;
  then consider a1,b1 such that
A14: p = a1*u + b1*v by A2,A13,A4,Th6;
A15: not are_Prop w,u by A1,Th12;
  not are_Prop v,w by A1,Th12;
  then consider a2,b2 such that
A16: q = a2*v + b2*w by A3,A4,A5,Th6;
A17: c*p + d*q = (c*a1)*u + (c*b1 + d*a2)*v + (d*b2)*w
  proof
    thus c*p + d*q = (c*a1)*u + (c*b1)*v + d*(a2*v + b2*w) by A14,A16,Lm7
      .= (c*a1)*u + (c*b1)*v + ((d*a2)*v + (d*b2)*w) by Lm7
      .= (c*a1)*u + (c*b1)*v + (d*a2)*v + (d*b2)*w by RLVECT_1:def 3
      .= (c*a1)*u + ((c*b1)*v + (d*a2)*v) + (d*b2)*w by RLVECT_1:def 3
      .= (c*a1)*u + (c*b1 + d*a2)*v + (d*b2)*w by RLVECT_1:def 6;
  end;
A18: now
    assume that
A19: not are_Prop p,q and
A20: p is not zero and
A21: q is not zero and
A22: b1 <> 0;
A23: now
      set c =1,d=-(b1*a2");
      set y=c*p + d*q;
      assume
A24:  a2<>0;
      then a2"<>0 by XCMPLX_1:202;
      then
A25:  b1*a2" <>0 by A22,XCMPLX_1:6;
A26:  y is not zero
      proof
        assume not y is not zero;
        then 0.V = 1*p + (-(b1*a2"))*q
          .= 1*p + (b1*a2")*(-q) by RLVECT_1:24
          .= 1*p + -((b1*a2")*q) by RLVECT_1:25;
        then -1*p = -((b1*a2")*q) by RLVECT_1:def 10;
        then 1*p = (b1*a2")*q by RLVECT_1:18;
        hence contradiction by A19,A25;
      end;
      c*b1 + d*a2 = b1 + (-b1)*(a2"*a2) .= b1 + (-b1)*1 by A24,XCMPLX_0:def 7
        .= 0;
      then y = (c*a1)*u + 0*v + (d*b2)*w by A17
        .= (c*a1)*u + 0.V + (d*b2)*w by RLVECT_1:10
        .= (c*a1)*u + (d*b2)*w;
      then
A27:  u,w,y are_LinDep by A15,A13,A5,Th6;
      p,q,y are_LinDep by A19,A20,A21,Th6;
      hence thesis by A26,A27;
    end;
    now
      set c =0,d=1;
      set y=c*p + d*q;
A28:  y = 0.V + 1*q by RLVECT_1:10
        .= 0.V + q by RLVECT_1:def 8
        .= q;
      assume a2=0;
      then c*b1 + d*a2 = 0;
      then y = (c*a1)*u + 0*v + (d*b2)*w by A17
        .= (c*a1)*u + 0.V + (d*b2)*w by RLVECT_1:10
        .= (c*a1)*u + (d*b2)*w;
      then
A29:  u,w,y are_LinDep by A15,A13,A5,Th6;
      p,q,y are_LinDep by A19,A20,A21,Th6;
      hence thesis by A21,A28,A29;
    end;
    hence thesis by A23;
  end;
  now
    assume that
A30: not are_Prop p,q and
A31: p is not zero and
A32: q is not zero and
A33: b1=0;
    now
      set c =1,d=0;
      set y=c*p + d*q;
A34:  y = p + 0*q by RLVECT_1:def 8
        .= p+0.V by RLVECT_1:10
        .= p;
      c*b1 + d*a2 = 0 by A33;
      then y = (c*a1)*u + 0*v + (d*b2)*w by A17
        .= (c*a1)*u + 0.V + (d*b2)*w by RLVECT_1:10
        .= (c*a1)*u + (d*b2)*w;
      then
A35:  u,w,y are_LinDep by A15,A13,A5,Th6;
      p,q,y are_LinDep by A30,A31,A32,Th6;
      hence thesis by A31,A34,A35;
    end;
    hence thesis;
  end;
  hence thesis by A6,A18;
end;
