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 Th14:
  a=b or a=c or b=c or a=d or b=d or d=c implies a,b,c,d are_coplanar
proof
A1: now
    assume a=b or a=c or b=c;
    then a,b,c are_collinear by ANPROJ_2:def 7;
    hence thesis by Th6;
  end;
A2: now
    assume a=d or b=d;
    then d,a,b are_collinear by ANPROJ_2:def 7;
    hence thesis by Th6;
  end;
A3: now
    assume d=c;
    then b,c,d are_collinear by ANPROJ_2:def 7;
    hence thesis by Th6;
  end;
  assume a=b or a=c or b=c or a=d or b=d or d=c;
  hence thesis by A1,A2,A3;
end;
