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
  not p,q,r are_collinear & a,b,c,p are_coplanar & a,b,c,r are_coplanar &
  a,b,c,q are_coplanar & p,q,r,s are_coplanar implies a,b,c,s are_coplanar
proof
  assume that
A1: not p,q,r are_collinear and
A2: a,b,c,p are_coplanar and
A3: a,b,c,r are_coplanar and
A4: a,b,c,q are_coplanar and
A5: p,q,r,s are_coplanar;
  now
    b,b,a are_collinear by ANPROJ_2:def 7;
    then
A6: b,a,c,b are_coplanar by Th6;
A7: b,a,c,r are_coplanar by A3,Th7;
    assume
A8: not a,b,c are_collinear;
    then
A9: not b,a,c are_collinear by Th1;
    a,p,q,r are_coplanar by A2,A3,A4,A8,Lm5;
    then
A10: p,q,r,a are_coplanar by Th7;
A11: c,a,b,r are_coplanar by A3,Th7;
A12: not c,a,b are_collinear by A8,Th1;
    c,a,b,p are_coplanar & c,a,b,q are_coplanar by A2,A4,Th7;
    then c,p,q,r are_coplanar by A12,A11,Lm5;
    then
A13: p,q,r,c are_coplanar by Th7;
    b,a,c,p are_coplanar & b,a,c,q are_coplanar by A2,A4,Th7;
    then p,q,r,b are_coplanar by A9,A6,A7,Th8;
    hence thesis by A1,A5,A10,A13,Th8;
  end;
  hence thesis by Th6;
end;
