let S, T be non empty TopSpace; for s1, s2, s3 being Point of S
for t1, t2, t3 being Point of T
for l1 being Path of [s1,t1],[s2,t2]
for l2 being Path of [s2,t2],[s3,t3]
for p1 being Path of s1,s2
for p2 being Path of s2,s3 st [s1,t1],[s2,t2] are_connected & [s2,t2],[s3,t3] are_connected & p1 = pr1 l1 & p2 = pr1 l2 holds
pr1 (l1 + l2) = p1 + p2
let s1, s2, s3 be Point of S; for t1, t2, t3 being Point of T
for l1 being Path of [s1,t1],[s2,t2]
for l2 being Path of [s2,t2],[s3,t3]
for p1 being Path of s1,s2
for p2 being Path of s2,s3 st [s1,t1],[s2,t2] are_connected & [s2,t2],[s3,t3] are_connected & p1 = pr1 l1 & p2 = pr1 l2 holds
pr1 (l1 + l2) = p1 + p2
let t1, t2, t3 be Point of T; for l1 being Path of [s1,t1],[s2,t2]
for l2 being Path of [s2,t2],[s3,t3]
for p1 being Path of s1,s2
for p2 being Path of s2,s3 st [s1,t1],[s2,t2] are_connected & [s2,t2],[s3,t3] are_connected & p1 = pr1 l1 & p2 = pr1 l2 holds
pr1 (l1 + l2) = p1 + p2
let l1 be Path of [s1,t1],[s2,t2]; for l2 being Path of [s2,t2],[s3,t3]
for p1 being Path of s1,s2
for p2 being Path of s2,s3 st [s1,t1],[s2,t2] are_connected & [s2,t2],[s3,t3] are_connected & p1 = pr1 l1 & p2 = pr1 l2 holds
pr1 (l1 + l2) = p1 + p2
let l2 be Path of [s2,t2],[s3,t3]; for p1 being Path of s1,s2
for p2 being Path of s2,s3 st [s1,t1],[s2,t2] are_connected & [s2,t2],[s3,t3] are_connected & p1 = pr1 l1 & p2 = pr1 l2 holds
pr1 (l1 + l2) = p1 + p2
let p1 be Path of s1,s2; for p2 being Path of s2,s3 st [s1,t1],[s2,t2] are_connected & [s2,t2],[s3,t3] are_connected & p1 = pr1 l1 & p2 = pr1 l2 holds
pr1 (l1 + l2) = p1 + p2
let p2 be Path of s2,s3; ( [s1,t1],[s2,t2] are_connected & [s2,t2],[s3,t3] are_connected & p1 = pr1 l1 & p2 = pr1 l2 implies pr1 (l1 + l2) = p1 + p2 )
assume that
A1:
( [s1,t1],[s2,t2] are_connected & [s2,t2],[s3,t3] are_connected )
and
A2:
p1 = pr1 l1
and
A3:
p2 = pr1 l2
; pr1 (l1 + l2) = p1 + p2
A4:
( s1,s2 are_connected & s2,s3 are_connected )
by A1, Th11;
hence
pr1 (l1 + l2) = p1 + p2
; verum