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 Th3:
  (for w ex a,b,c st w = a*p + b*q + c*r) & (for a,b,c st a*p + b*q
  + c*r = 0.V holds a = 0 & b = 0 & c = 0) implies for u,u1 ex y st p,q,y
  are_LinDep & u,u1,y are_LinDep & y is not zero
proof
  assume that
A1: for w ex a,b,c st w = a*p + b*q + c*r and
A2: for a,b,c st a*p + b*q + c*r = 0.V holds a = 0 & b = 0 & c = 0;
  let u,u1;
  consider a,b,c such that
A3: u = a*p + b*q + c*r by A1;
  consider a1,b1,c1 such that
A4: u1 = a1*p + b1*q + c1*r by A1;
A5: a3*u + b3*u1 = (a3*a + b3*a1)*p + (a3*b + b3*b1)*q + (a3*c + b3*c1)*r
  proof
    a3*u = (a3*a)*p + (a3*b)*q + (a3*c)*r by A3,Lm2;
    hence
    a3*u + b3*u1 = ((a3*a)*p + (a3*b)*q + (a3*c)*r) + ((b3*a1)*p + (b3*b1
    )*q + (b3*c1)*r) by A4,Lm2
      .= ((a3*a)*p + (b3*a1)*p) + ((a3*b)*q + (b3*b1)*q) + ((a3*c)*r + (b3*
    c1)*r) by Lm3
      .= (a3*a + b3*a1)*p + ((a3*b)*q + (b3*b1)*q) + ((a3*c)*r + (b3*c1)*r)
    by RLVECT_1:def 6
      .= (a3*a + b3*a1)*p + (a3*b + b3*b1)*q + ((a3*c)*r + (b3*c1)*r) by
RLVECT_1:def 6
      .= (a3*a + b3*a1)*p + (a3*b + b3*b1)*q + (a3*c + b3*c1)*r by
RLVECT_1:def 6;
  end;
A6: q is not zero by A2,Th1;
A7: now
A8: now
      assume not u1 is not zero;
      then u1 = 0.V;
      then p,q,q are_LinDep & u,u1,q are_LinDep by ANPROJ_1:10,11;
      hence thesis by A6;
    end;
A9: now
      assume not u is not zero;
      then u = 0.V;
      then p,q,q are_LinDep & u,u1,q are_LinDep by ANPROJ_1:10,11;
      hence thesis by A6;
    end;
A10: now
      assume are_Prop u,u1;
      then p,q,q are_LinDep & u,u1,q are_LinDep by ANPROJ_1:11;
      hence thesis by A6;
    end;
    assume are_Prop u,u1 or not u is not zero or not u1 is not zero;
    hence thesis by A10,A9,A8;
  end;
A11: p is not zero & not are_Prop p,q by A2,Th1;
A12: now
    assume that
A13: not are_Prop u,u1 and
A14: u is not zero and
A15: u1 is not zero and
A16: c <> 0;
A17: now
      set a3 = 1,b3 = -(c*c1");
      set y = a3*u + b3*u1;
      assume
A18:  c1 <> 0;
      then c1"<>0 by XCMPLX_1:202;
      then
A19:  c*c1" <> 0 by A16,XCMPLX_1:6;
A20:  y is not zero
      proof
        assume not y is not zero;
        then 0.V = 1*u + (-(c*c1"))*u1
          .= 1*u + (c*c1")*(-u1) by RLVECT_1:24
          .= 1*u + -((c*c1")*u1) by RLVECT_1:25;
        then -1*u = -((c*c1")*u1) by RLVECT_1:def 10;
        then 1*u = (c*c1")*u1 by RLVECT_1:18;
        hence contradiction by A13,A19;
      end;
      a3*c + b3*c1 = c + (-c)*(c1"*c1) .= c + (-c)*1 by A18,XCMPLX_0:def 7
        .= 0;
      then y = (a3*a + b3*a1)*p + (a3*b + b3*b1)*q + 0*r by A5
        .= (a3*a + b3*a1)*p + (a3*b + b3*b1)*q + 0.V by RLVECT_1:10
        .= (a3*a + b3*a1)*p + (a3*b + b3*b1)*q by RLVECT_1:4;
      then
A21:  p,q,y are_LinDep by A6,A11,ANPROJ_1:6;
      u,u1,y are_LinDep by A13,A14,A15,ANPROJ_1:6;
      hence thesis by A20,A21;
    end;
    now
      set a3 = 0,b3 = 1;
      set y = a3*u + b3*u1;
A22:  y = 0*u + u1 by RLVECT_1:def 8
        .= 0.V + u1 by RLVECT_1:10
        .= u1 by RLVECT_1:4;
      assume c1 = 0;
      then a3*c + b3*c1 = 0;
      then y = (a3*a + b3*a1)*p + (a3*b + b3*b1)*q + 0*r by A5
        .= (a3*a + b3*a1)*p + (a3*b + b3*b1)*q + 0.V by RLVECT_1:10
        .= (a3*a + b3*a1)*p + (a3*b + b3*b1)*q by RLVECT_1:4;
      then
A23:  p,q,y are_LinDep by A6,A11,ANPROJ_1:6;
      u,u1,y are_LinDep by A13,A14,A15,ANPROJ_1:6;
      hence thesis by A15,A22,A23;
    end;
    hence thesis by A17;
  end;
  now
    assume that
A24: not are_Prop u,u1 and
A25: u is not zero and
A26: u1 is not zero and
A27: c = 0;
    now
      set a3 = 1,b3 = 0;
      set y = a3*u + b3*u1;
A28:  y = u + 0*u1 by RLVECT_1:def 8
        .= u + 0.V by RLVECT_1:10
        .= u by RLVECT_1:4;
      a3*c + b3*c1 = 0 by A27;
      then y = (a3*a + b3*a1)*p + (a3*b + b3*b1)*q + 0*r by A5
        .= (a3*a + b3*a1)*p + (a3*b + b3*b1)*q + 0.V by RLVECT_1:10
        .= (a3*a + b3*a1)*p + (a3*b + b3*b1)*q by RLVECT_1:4;
      then
A29:  p,q,y are_LinDep by A6,A11,ANPROJ_1:6;
      u,u1,y are_LinDep by A24,A25,A26,ANPROJ_1:6;
      hence thesis by A25,A28,A29;
    end;
    hence thesis;
  end;
  hence thesis by A7,A12;
end;
