let FCPS be up-3-dimensional CollProjectiveSpace; :: thesis: for a, b, c, b9 being Element of FCPS st not a,b,c is_collinear & a,b,b9 is_collinear & a <> b9 holds
not a,b9,c is_collinear
let a, b, c, b9 be Element of FCPS; :: thesis: ( not a,b,c is_collinear & a,b,b9 is_collinear & a <> b9 implies not a,b9,c is_collinear )
assume that
A1: not a,b,c is_collinear and
A2: a,b,b9 is_collinear and
A3: a <> b9 ; :: thesis: not a,b9,c is_collinear
assume A4: a,b9,c is_collinear ; :: thesis: contradiction
a,b9,b is_collinear by A2, Th1;
hence contradiction by A1, A3, A4, COLLSP:11; :: thesis: verum