let M be MidSp; :: thesis: for a being Element of M
for u being Vector of M ex b being Element of M st u = [a,b] ~
let a be Element of M; :: thesis: for u being Vector of M ex b being Element of M st u = [a,b] ~
let u be Vector of M; :: thesis: ex b being Element of M st u = [a,b] ~
consider p being Element of [: the carrier of M, the carrier of M:] such that
A1: u = p ~ by Def8;
consider b being Element of M such that
A2: p `1 ,p `2 @@ a,b by Th26;
[(p `1),(p `2)] ## [a,b] by A2, Th31;
then p ## [a,b] by MCART_1:21;
then p ~ = [a,b] ~ by Th42;
hence ex b being Element of M st u = [a,b] ~ by A1; :: thesis: verum