reserve V for RealLinearSpace;
reserve p,q,u,v,w,y for VECTOR of V;
reserve a,b,c,d for Real;

theorem Th21:
  (ex p,q st (for w ex a,b st a*p + b*q=w)) implies for u,v,w,y st
not u,v // w,y & not u,v // y,w ex z being VECTOR of V st (u,v // u,z or u,v //
  z,u) & (w,y // w,z or w,y // z,w)
proof
  given p,q such that
A1: for w ex a,b st a*p + b*q=w;
  let u,v,w,y such that
A2: not u,v // w,y and
A3: not u,v // y,w;
  consider r1,s1 being Real such that
A4: r1*p + s1*q = v-u by A1;
  consider r2,s2 being Real such that
A5: r2*p + s2*q = y-w by A1;
  set r = r1*s2 - r2*s1;
A6: now
    assume
A7: r = 0;
A8: now
      assume that
A9:   r1<>0 and
A10:  r2=0;
      s2<>0
      proof
        assume s2=0;
        then y-w = 0.V + 0*q by A5,A10,RLVECT_1:10
          .= 0.V + 0.V by RLVECT_1:10
          .= 0.V by RLVECT_1:4;
        then y=w by RLVECT_1:21;
        hence contradiction by A2;
      end;
      hence contradiction by A7,A9,A10,XCMPLX_1:6;
    end;
A11: now
      assume
A12:  r1=0;
A13:  s1<>0
      proof
        assume s1=0;
        then v-u = 0.V + 0*q by A4,A12,RLVECT_1:10
          .= 0.V + 0.V by RLVECT_1:10
          .= 0.V by RLVECT_1:4;
        then u=v by RLVECT_1:21;
        hence contradiction by A2;
      end;
      then
A14:  r2=0 by A7,A12,XCMPLX_1:6;
A15:  s2<>0
      proof
        assume s2=0;
        then y-w = 0.V + 0*q by A5,A14,RLVECT_1:10
          .= 0.V + 0.V by RLVECT_1:10
          .= 0.V by RLVECT_1:4;
        then y=w by RLVECT_1:21;
        hence contradiction by A2;
      end;
      y-w = 0.V + s2*q by A5,A14,RLVECT_1:10
        .= s2*q by RLVECT_1:4;
      then
A16:  (s2)"*(y-w) = ((s2)"*s2)*q by RLVECT_1:def 7
        .= 1*q by A15,XCMPLX_0:def 7
        .= q by RLVECT_1:def 8;
      v-u = 0.V + s1*q by A4,A12,RLVECT_1:10
        .= s1*q by RLVECT_1:4;
      then
A17:  (s1)"*(v-u) = ((s1)"*s1)*q by RLVECT_1:def 7
        .= 1*q by A13,XCMPLX_0:def 7
        .= q by RLVECT_1:def 8;
      s1"<>0 by A13,XCMPLX_1:202;
      hence contradiction by A2,A3,A17,A16,Lm1;
    end;
A18: now
      assume that
A19:  r1<>0 and
A20:  r2<>0 and
A21:  s1 = 0;
      v-u = r1*p + 0.V by A4,A21,RLVECT_1:10
        .= r1*p by RLVECT_1:4;
      then
A22:  (r1)"*(v-u) = ((r1)"*r1)*p by RLVECT_1:def 7
        .= 1*p by A19,XCMPLX_0:def 7
        .= p by RLVECT_1:def 8;
      s2 = 0 by A7,A19,A21,XCMPLX_1:6;
      then y-w = r2*p + 0.V by A5,RLVECT_1:10
        .= r2*p by RLVECT_1:4;
      then
A23:  (r2)"*(y-w) = ((r2)"*r2)*p by RLVECT_1:def 7
        .= 1*p by A20,XCMPLX_0:def 7
        .= p by RLVECT_1:def 8;
      r1"<>0 by A19,XCMPLX_1:202;
      hence contradiction by A2,A3,A22,A23,Lm1;
    end;
    now
      assume that
A24:  r1<>0 and
      r2<>0 and
      s1<>0 and
      s2<>0;
      r2*(v-u) = r2*(r1*p) + r2*(s1*q) by A4,RLVECT_1:def 5
        .=(r2*r1)*p + r2*(s1*q) by RLVECT_1:def 7
        .= (r1*r2)*p + (r1*s2)*q by A7,RLVECT_1:def 7
        .= r1*(r2*p) + (r1*s2)*q by RLVECT_1:def 7
        .= r1*(r2*p) + r1*(s2*q) by RLVECT_1:def 7
        .= r1*(y-w) by A5,RLVECT_1:def 5;
      hence contradiction by A2,A3,A24,Lm1;
    end;
    hence contradiction by A7,A11,A8,A18,XCMPLX_1:6;
  end;
  consider r3,s3 being Real such that
A25: r3*p + s3*q = u-w by A1;
  set a= r2*s3 - r3*s2, b= r1*s3 - r3*s1;
A26: b*r2 = r1*a + r3*r;
  set z = u + (r"*a)*(v-u);
A27: r*(z-u) = r*z - r*u by RLVECT_1:34
    .= r*u + r*((r"*a)*(v-u)) - r*u by RLVECT_1:def 5
    .= r*u + (r*(r"*a))*(v-u) - r*u by RLVECT_1:def 7
    .= r*u + ((r*r")*a)*(v-u) - r*u
    .= r*u + (1*a)*(v-u) - r*u by A6,XCMPLX_0:def 7
    .= a*(v-u) + (r*u - r*u) by RLVECT_1:def 3
    .= a*(v-u) + 0.V by RLVECT_1:15
    .= a*(v-u) by RLVECT_1:4;
A28: r*(z-w) = r*z - r*w by RLVECT_1:34
    .= r*u + r*((r"*a)*(v-u)) - r*w by RLVECT_1:def 5
    .= r*u + (r*(r"*a))*(v-u) - r*w by RLVECT_1:def 7
    .= r*u + ((r*r")*a)*(v-u) - r*w
    .= r*u + (1*a)*(v-u) - r*w by A6,XCMPLX_0:def 7
    .= a*(v-u) + (r*u - r*w) by RLVECT_1:def 3
    .= a*(r1*p + s1*q) + r*(r3*p + s3*q) by A4,A25,RLVECT_1:34
    .= a*(r1*p) + a*(s1*q) + r*(r3*p + s3*q) by RLVECT_1:def 5
    .= a*(r1*p) + a*(s1*q) + (r*(r3*p) + r*(s3*q)) by RLVECT_1:def 5
    .= (a*r1)*p + a*(s1*q) + (r*(r3*p) + r*(s3*q)) by RLVECT_1:def 7
    .= (a*r1)*p + (a*s1)*q + (r*(r3*p) + r*(s3*q)) by RLVECT_1:def 7
    .= (a*r1)*p + (a*s1)*q + ((r*r3)*p + r*(s3*q)) by RLVECT_1:def 7
    .= (a*r1)*p + (a*s1)*q + ((r*s3)*q + (r*r3)*p) by RLVECT_1:def 7
    .= (a*r1)*p + (a*s1)*q + (r*s3)*q + (r*r3)*p by RLVECT_1:def 3
    .= ((a*s1)*q + (r*s3)*q) + (a*r1)*p + (r*r3)*p by RLVECT_1:def 3
    .= ((a*s1)*q + (r*s3)*q) + ((a*r1)*p + (r*r3)*p) by RLVECT_1:def 3
    .= (a*s1 + r*s3)*q + ((a*r1)*p + (r*r3)*p) by RLVECT_1:def 6
    .= (b*s2)*q + (b*r2)*p by A26,RLVECT_1:def 6
    .= b*(s2*q) + (b*r2)*p by RLVECT_1:def 7
    .= b*(s2*q) + b*(r2*p) by RLVECT_1:def 7
    .= b*(y-w) by A5,RLVECT_1:def 5;
A29: b*s2 = s1*a + s3*r;
   per cases;
   suppose that
A30: a=0 and
A31: b<>0;
    r*(z-u)=0.V by A27,A30,RLVECT_1:10;
    then z-u=0.V by A6,RLVECT_1:11;
    then z=u by RLVECT_1:21;
    then
A32: u,v // u,z;
    w,y // w,z or w,y // z,w by A28,A31,Lm1;
    hence thesis by A32;
  end;
  suppose a=0 & b=0;
    then r3=0 & s3=0 by A6,A26,A29,XCMPLX_1:6;
    then 0.V + 0*q = u-w by A25,RLVECT_1:10;
    then 0.V + 0.V = u-w by RLVECT_1:10;
    then 0.V=u-w by RLVECT_1:4;
    then u=w by RLVECT_1:21;
    then
A33: w,y // w,u;
    u,v // u,u;
    hence thesis by A33;
  end;
  suppose that
A34: a<>0 and
A35: b=0;
    r*(z-w)=0.V by A28,A35,RLVECT_1:10;
    then z-w=0.V by A6,RLVECT_1:11;
    then z=w by RLVECT_1:21;
    then
A36: w,y // w,z;
    u,v // u,z or u,v // z,u by A27,A34,Lm1;
    hence thesis by A36;
  end;
  suppose that
A37: a<>0 and
A38: b<>0;
A39: w,y // w,z or w,y // z,w by A28,A38,Lm1;
    u,v // u,z or u,v // z,u by A27,A37,Lm1;
    hence thesis by A39;
  end;
end;
