let PS be ParSp; :: thesis: for a, b, c, p, r being Element of PS st not a,b '||' a,c & a,c '||' p,r & b,c '||' p,r holds
p = r
let a, b, c, p, r be Element of PS; :: thesis: ( not a,b '||' a,c & a,c '||' p,r & b,c '||' p,r implies p = r )
assume that
A1: ( not a,b '||' a,c & a,c '||' p,r ) and
A2: b,c '||' p,r ; :: thesis: p = r
A3: p,r '||' b,c by A2, Th23;
( not a,c '||' b,c & p,r '||' a,c ) by A1, Th23, Th29;
hence p = r by A3, Def11; :: thesis: verum