reserve FCPS for up-3-dimensional CollProjectiveSpace;
reserve a,a9,b,b9,c,c9,d,d9,o,p,q,r,s,t,u,x,y,z for Element of FCPS;

theorem Th8:
  not a,b,c are_collinear & a,b,c,p are_coplanar & a,b,c,q
  are_coplanar & a,b,c,r are_coplanar & a,b,c,s are_coplanar implies
  p,q,r,s are_coplanar
proof
  assume that
A1: not a,b,c are_collinear and
A2: a,b,c,p are_coplanar and
A3: a,b,c,q are_coplanar and
A4: a,b,c,r are_coplanar and
A5: a,b,c,s are_coplanar;
A6: a,b,p,q are_coplanar by A1,A2,A3,Lm4;
A7: a,b,q,r are_coplanar by A1,A3,A4,Lm4;
A8: a,b,p,r are_coplanar by A1,A2,A4,Lm4;
A9: a,b,q,s are_coplanar by A1,A3,A5,Lm4;
A10: now
A11: q,a,b,r are_coplanar by A7,Th7;
    assume
A12: not a,b,q are_collinear;
    then
A13: not q,a,b are_collinear by Th1;
A14: q,a,b,p are_coplanar by A6,Th7;
    then
A15: q,a,p,r are_coplanar by A13,A11,Lm4;
A16: q,a,b,s are_coplanar by A9,Th7;
    then
A17: q,a,p,s are_coplanar by A13,A14,Lm4;
A18: now
      assume not q,a,p are_collinear;
      then
A19:  not q,p,a are_collinear by Th1;
      q,p,a,r are_coplanar & q,p,a,s are_coplanar by A15,A17,Th7;
      then q,p,r,s are_coplanar by A19,Lm4;
      hence thesis by Th7;
    end;
A20: q,a,r,s are_coplanar by A13,A11,A16,Lm4;
A21: now
      assume not q,a,r are_collinear;
      then
A22:  not q,r,a are_collinear by Th1;
      q,r,a,p are_coplanar & q,r,a,s are_coplanar by A15,A20,Th7;
      then q,r,p,s are_coplanar by A22,Lm4;
      hence thesis by Th7;
    end;
A23: q<>a by A12,ANPROJ_2:def 7;
    now
      assume q,a,p are_collinear & q,a,r are_collinear;
      then q,p,r are_collinear by A23,COLLSP:6;
      then p,q,r are_collinear by Th1;
      hence thesis by Th6;
    end;
    hence thesis by A18,A21;
  end;
A24: a,b,r,s are_coplanar by A1,A4,A5,Lm4;
A25: now
A26: r,a,b,q are_coplanar by A7,Th7;
    assume
A27: not a,b,r are_collinear;
    then
A28: not r,a,b are_collinear by Th1;
A29: r,a,b,p are_coplanar by A8,Th7;
    then
A30: r,a,p,q are_coplanar by A28,A26,Lm4;
A31: r,a,b,s are_coplanar by A24,Th7;
    then
A32: r,a,p,s are_coplanar by A28,A29,Lm4;
A33: now
      assume not r,a,p are_collinear;
      then
A34:  not r,p,a are_collinear by Th1;
      r,p,a,q are_coplanar & r,p,a,s are_coplanar by A30,A32,Th7;
      then r,p,q,s are_coplanar by A34,Lm4;
      hence thesis by Th7;
    end;
A35: r,a,q,s are_coplanar by A28,A26,A31,Lm4;
A36: now
      assume not r,a,q are_collinear;
      then
A37:  not r,q,a are_collinear by Th1;
      r,q,a,p are_coplanar & r,q,a,s are_coplanar by A30,A35,Th7;
      then r,q,p,s are_coplanar by A37,Lm4;
      hence thesis by Th7;
    end;
A38: r<>a by A27,ANPROJ_2:def 7;
    now
      assume r,a,p are_collinear & r,a,q are_collinear;
      then r,p,q are_collinear by A38,COLLSP:6;
      then p,q,r are_collinear by Th1;
      hence thesis by Th6;
    end;
    hence thesis by A33,A36;
  end;
A39: a,b,p,s are_coplanar by A1,A2,A5,Lm4;
A40: now
A41: p,a,b,r are_coplanar by A8,Th7;
    assume
A42: not a,b,p are_collinear;
    then
A43: not p,a,b are_collinear by Th1;
A44: p,a,b,q are_coplanar by A6,Th7;
    then
A45: p,a,q,r are_coplanar by A43,A41,Lm4;
A46: p,a,b,s are_coplanar by A39,Th7;
    then
A47: p,a,q,s are_coplanar by A43,A44,Lm4;
A48: now
      assume not p,a,q are_collinear;
      then
A49:  not p,q,a are_collinear by Th1;
      p,q,a,r are_coplanar & p,q,a,s are_coplanar by A45,A47,Th7;
      hence thesis by A49,Lm4;
    end;
A50: p,a,r,s are_coplanar by A43,A41,A46,Lm4;
A51: now
      assume not p,a,r are_collinear;
      then
A52:  not p,r,a are_collinear by Th1;
      p,r,a,q are_coplanar & p,r,a,s are_coplanar by A45,A50,Th7;
      then p,r,q,s are_coplanar by A52,Lm4;
      hence thesis by Th7;
    end;
A53: p<>a by A42,ANPROJ_2:def 7;
    now
      assume p,a,q are_collinear & p,a,r are_collinear;
      then p,q,r are_collinear by A53,COLLSP:6;
      hence thesis by Th6;
    end;
    hence thesis by A48,A51;
  end;
A54: a<>b by A1,ANPROJ_2:def 7;
  now
    assume a,b,p are_collinear & a,b,q are_collinear & a,b,r are_collinear;
    then p,q,r are_collinear by A54,ANPROJ_2:def 8;
    hence thesis by Th6;
  end;
  hence thesis by A40,A10,A25;
end;
