let FCPS be up-3-dimensional CollProjectiveSpace; :: thesis: for a, b, c, d being Element of FCPS st a,b,c,d are_coplanar holds

b,a,c,d are_coplanar

let a, b, c, d be Element of FCPS; :: thesis: ( a,b,c,d are_coplanar implies b,a,c,d are_coplanar )

assume a,b,c,d are_coplanar ; :: thesis: b,a,c,d are_coplanar

then consider x being Element of FCPS such that

A1: a,b,x are_collinear and

A2: c,d,x are_collinear ;

b,a,x are_collinear by A1, Th1;

hence b,a,c,d are_coplanar by A2; :: thesis: verum

b,a,c,d are_coplanar

let a, b, c, d be Element of FCPS; :: thesis: ( a,b,c,d are_coplanar implies b,a,c,d are_coplanar )

assume a,b,c,d are_coplanar ; :: thesis: b,a,c,d are_coplanar

then consider x being Element of FCPS such that

A1: a,b,x are_collinear and

A2: c,d,x are_collinear ;

b,a,x are_collinear by A1, Th1;

hence b,a,c,d are_coplanar by A2; :: thesis: verum