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 Th7:
  a,b,c,d are_coplanar implies b,c,d,a are_coplanar & c,d,a,b
  are_coplanar & d,a,b,c are_coplanar & b,a,c,d are_coplanar & c,b,d,a
  are_coplanar & d,c,a,b are_coplanar & a,d,b,c are_coplanar & a,c,d,b
  are_coplanar & b,d,a,c are_coplanar & c,a,b,d are_coplanar & d,b,c,a
  are_coplanar & c,a,d,b are_coplanar & d,b,a,c are_coplanar & a,c,b,d
  are_coplanar & b,d,c,a are_coplanar & a,b,d,c are_coplanar & a,d,c,b
  are_coplanar & b,c,a,d are_coplanar & b,a,d,c are_coplanar & c,b,a,d
  are_coplanar & c,d,b,a are_coplanar & d,a,c,b are_coplanar & d,c,b,a
  are_coplanar
proof
  assume
A1: a,b,c,d are_coplanar;
  then
 b,a,c,d are_coplanar by Lm2;
  then a,c,d,b are_coplanar by Lm3;
  then
A2: c,d,b,a are_coplanar by Lm3;
  then
A3: d,b,a,c are_coplanar by Lm3;
  then b,d,a,c are_coplanar by Lm2;
  then
A4: d,a,c,b are_coplanar by Lm3;
  then
 a,c,b,d are_coplanar by Lm3;
  then c,a,b,d are_coplanar by Lm2;
  then
A5: a,b,d,c are_coplanar by Lm3;
  then
A6: b,d,c,a are_coplanar by Lm3;
  then
 d,c,a,b are_coplanar by Lm3;
  then c,d,a,b are_coplanar by Lm2;
  then
A7: d,a,b,c are_coplanar by Lm3;
  then a,d,b,c are_coplanar by Lm2;
  then d,b,c,a are_coplanar by Lm3;
  then b,c,a,d are_coplanar by Lm3;
  hence thesis by A1,A2,A3,A4,A5,A6,A7,Lm2,Lm3;
end;
