let T be non empty arcwise_connected TopSpace; :: thesis: for a, b, c, d being Point of T
for f being Path of a,b
for g being Path of b,c
for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let a, b, c, d be Point of T; :: thesis: for f being Path of a,b
for g being Path of b,c
for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let f be Path of a,b; :: thesis: for g being Path of b,c
for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let g be Path of b,c; :: thesis: for h being Path of c,d holds rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
let h be Path of c,d; :: thesis: rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h)
( a,b are_connected & b,c are_connected & c,d are_connected ) by BORSUK_2:def 3;
hence rng ((f + g) + h) = ((rng f) \/ (rng g)) \/ (rng h) by Th39; :: thesis: verum