
theorem Th43:
  for P,Q,R being non point_at_infty Point of ProjectiveSpace TOP-REAL 3
  for u,v,w being non zero Element of TOP-REAL 3 st
  P = Dir u & Q = Dir v & R = Dir w &
  u`3 = 1 & v`3 = 1 & w`3 = 1 holds
  P,Q,R are_collinear iff u,v,w are_collinear
  proof
    let P,Q,R be non point_at_infty Point of ProjectiveSpace TOP-REAL 3;
    let u,v,w be non zero Element of TOP-REAL 3;
    assume that
A1: P = Dir u & Q = Dir v & R = Dir w and
A2: u`3 = 1 & v`3 = 1 & w`3 = 1;
    reconsider i = 3 as Nat;
    hereby
      assume P,Q,R are_collinear;
      then consider u9,v9,w9 be Element of TOP-REAL 3 such that
A3:   P = Dir u9 & Q = Dir v9 & R = Dir w9 &
      u9 is not zero & v9 is not zero & w9 is not zero and
A4:   ex a,b,c be Real st a*u9+b*v9+c*w9=0.TOP-REAL 3 &
      (a <> 0 or b <> 0 or c <> 0) by ANPROJ_8:11;
A5:   |{u9,v9,w9}| = 0 by A4,ANPROJ_8:41;
      are_Prop u,u9 by A1,A3,ANPROJ_1:22;
      then consider a be Real such that
A6:   a <> 0 and
A7:   u9 = a * u by ANPROJ_1:1;
      are_Prop v,v9 by A1,A3,ANPROJ_1:22;
      then consider b be Real such that
A8:   b <> 0 and
A9:   v9 = b * v by ANPROJ_1:1;
      are_Prop w,w9 by A1,A3,ANPROJ_1:22;
      then consider c be Real such that
A10:  c <> 0 and
A11:  w9 = c * w by ANPROJ_1:1;
      reconsider d = a * (b * c) as non zero Real by A6,A8,A10;
      0 = a * |{u,b * v,c * w}| by ANPROJ_8:31,A5,A7,A9,A11
       .= a * (b * |{u,v,c * w}|) by ANPROJ_8:32
       .= a * (b * (c * |{u,v,w}|)) by ANPROJ_8:33
       .= d * |{u,v,w}|;
      then |{u,v,w}| = 0;
      then consider  a,b,c be Real such that
A12:  a * u + b * v + c * w = 0.TOP-REAL 3 and
A13:  a <> 0 or b <> 0 or c <> 0 by ANPROJ_8:43,ANPROJ_1:def 2;
      reconsider aubv = a * u + b * v,cw = c * w,au = a * u,
        bv = b * v as Element of TOP-REAL 3;
A14:  cw is Element of REAL 3 by EUCLID:22;
A15:  aubv is Element of REAL 3 by EUCLID:22;
      au is Element of REAL 3 & bv is Element of REAL 3 by EUCLID:22;
      then
A16:  aubv.3 = au.3 + bv.3 by RVSUM_1:11
            .= a * u.3 + bv.3 by RVSUM_1:44
            .= a * u.3 + b * v.3 by RVSUM_1:44;
A17:  cw.3 = c * w.3 by RVSUM_1:44;
      |[(aubv + cw)`1,(aubv + cw)`2,(aubv+cw)`3]| = |[0,0,0]|
        by A12,EUCLID_5:3,4;
      then
A18:  0 = (aubv + cw)`3 by FINSEQ_1:78
       .= (aubv + cw).3 by EUCLID_5:def 3
       .= aubv.3+cw.3 by A14,A15,RVSUM_1:11
       .= a * u`3 + b * v.3 + c * w.3 by A16,A17,EUCLID_5:def 3
       .= a * u`3 + b * v`3 + c * w.3 by EUCLID_5:def 3
       .= a * u`3 + b * v`3 + c * w`3 by EUCLID_5:def 3
       .= a + b + c by A2;
      thus u,v,w are_collinear
      proof
        per cases by A13;
        suppose
A19:      a <> 0;
          reconsider L = Line(v,w) as line of TOP-REAL 3;
          u in L & v in L & w in L by A18,A12,Th42,A19,RLTOPSP1:72;
          hence thesis by RLTOPSP1:def 16;
        end;
        suppose
A20:      b <> 0;
A21:      b * v + c * w + a * u = 0.TOP-REAL 3 by A12,RVSUM_1:15;
A22:      b + c + a = 0 by A18;
          reconsider L = Line(w,u) as line of TOP-REAL 3;
          v in L & w in L & u in L by A22,A20,A21,Th42,RLTOPSP1:72;
          hence thesis by RLTOPSP1:def 16;
        end;
        suppose
A23:      c <> 0;
A24:      c * w + a * u + b * v = 0.TOP-REAL 3 by A12,RVSUM_1:15;
A25:      c + a + b = 0 by A18;
          reconsider L = Line(u,v) as line of TOP-REAL 3;
          w in L & u in L & v in L by A25,A23,A24,Th42,RLTOPSP1:72;
          hence thesis by RLTOPSP1:def 16;
        end;
      end;
    end;
    assume u,v,w are_collinear;
    then per cases by TOPREAL9:67;
    suppose u in LSeg(v,w);
      then consider r be Real such that 0 <= r & r <= 1 and
A26:  u = (1 - r) * v + r * w by RLTOPSP1:76;
      reconsider a = 1,b = r - 1,c = -r as Real;
      1 * u + (r - 1) * v + (-r) * w = 0.TOP-REAL 3
      proof
        reconsider vw = v + w as Element of REAL 3 by EUCLID:22;
        1 * u = (1 - r) * v + r * w by A26,RVSUM_1:52;
        then 1 * u + (r - 1) * v + (-r) * w
          = r * w + ((1 - r) * v + (r - 1) * v) + (-r) * w by RVSUM_1:15
         .= r * w + ((1 - r) + (r - 1)) * v + (-r) * w by RVSUM_1:50
         .= 0 * v + (r * w + (-r) * w) by RVSUM_1:15
         .= 0 * v + (r + (-r)) * w by RVSUM_1:50
         .= 0 * vw by RVSUM_1:51
         .= i|->0 by RVSUM_1:53
         .= 0.TOP-REAL 3 by EUCLID_5:4,FINSEQ_2:62;
        hence thesis;
      end;
      hence thesis by A1,ANPROJ_8:11;
    end;
    suppose v in LSeg(w,u);
      then consider r be Real such that 0 <= r & r <= 1 and
A27:  v = (1 - r) * w + r * u by RLTOPSP1:76;
      reconsider a = -r,b = 1,c = r - 1 as Real;
      (-r) * u + 1 * v + (r - 1) * w = 0.TOP-REAL 3
      proof
        reconsider uw = u + w as Element of REAL 3 by EUCLID:22;
        1 * v = (1 - r) * w + r * u by A27,RVSUM_1:52;
        then (-r) * u + 1 * v + (r - 1) * w
          = (-r) * u + r * u + (1 - r) * w + (r - 1) * w by RVSUM_1:15
         .= ((-r) + r) * u + (1 - r) * w + (r - 1) * w by RVSUM_1:50
         .= 0 * u + ((1 - r) * w + (r - 1) * w) by RVSUM_1:15
         .= 0 * u + (((1 - r) + (r - 1)) * w) by RVSUM_1:50
         .= 0 * uw by RVSUM_1:51
         .= i|->0 by RVSUM_1:53
         .= 0.TOP-REAL 3 by FINSEQ_2:62,EUCLID_5:4;
        hence thesis;
      end;
      hence thesis by A1,ANPROJ_8:11;
    end;
    suppose w in LSeg(u,v);
      then consider r be Real such that 0 <= r & r <= 1 and
A28:  w = (1 - r) * u + r * v by RLTOPSP1:76;
      reconsider a = r - 1,b = -r,c = 1 as Real;
      (r - 1) * u + (-r) * v + 1 * w = 0.TOP-REAL 3
      proof
        reconsider vu = v + u as Element of REAL 3 by EUCLID:22;
        1 * w = (1 - r) * u + r * v by A28,RVSUM_1:52;
        then (r - 1) * u + (-r) * v + 1 * w
          = (r - 1) * u + ((-r) * v + (r * v + (1 - r) * u)) by RVSUM_1:15
         .= (r - 1) * u + ((-r) * v + r * v + (1 - r) * u) by RVSUM_1:15
         .= (r - 1) * u + (((-r) + r) * v + (1 - r) * u) by RVSUM_1:50
         .= 0 * v + ((r - 1) * u + (1 - r) * u) by RVSUM_1:15
         .= 0 * v + (((r - 1) + (1 - r)) * u) by RVSUM_1:50
         .= 0 * vu by RVSUM_1:51
         .= i|->0 by RVSUM_1:53
         .= 0.TOP-REAL 3 by FINSEQ_2:62,EUCLID_5:4;
        hence thesis;
      end;
      hence thesis by A1,ANPROJ_8:11;
    end;
  end;
