let FCPS be up-3-dimensional CollProjectiveSpace; for a, b, c, d being Element of FCPS st a,b,c,d are_coplanar holds
( 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 )
let a, b, c, d be Element of FCPS; ( 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 ) )
assume A1:
a,b,c,d are_coplanar
; ( 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 )
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
( 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 )
by A1, A2, A3, A4, A5, A6, A7, Lm2, Lm3; verum