let p be Point of (TOP-REAL 2); :: thesis: for Z being non empty Subset of (TOP-REAL 2) st p in N-most Z holds
( p `2 = (N-min Z) `2 & ( Z is compact implies ( (N-min Z) `1 <= p `1 & p `1 <= (N-max Z) `1 ) ) )
let Z be non empty Subset of (TOP-REAL 2); :: thesis: ( p in N-most Z implies ( p `2 = (N-min Z) `2 & ( Z is compact implies ( (N-min Z) `1 <= p `1 & p `1 <= (N-max Z) `1 ) ) ) )
A1: ( (NW-corner Z) `2 = N-bound Z & (N-min Z) `2 = N-bound Z ) by EUCLID:52;
A2: (NE-corner Z) `2 = N-bound Z by EUCLID:52;
assume A3: p in N-most Z ; :: thesis: ( p `2 = (N-min Z) `2 & ( Z is compact implies ( (N-min Z) `1 <= p `1 & p `1 <= (N-max Z) `1 ) ) )
then p in LSeg ((NW-corner Z),(NE-corner Z)) by XBOOLE_0:def 4;
hence p `2 = (N-min Z) `2 by A1, A2, GOBOARD7:6; :: thesis: ( Z is compact implies ( (N-min Z) `1 <= p `1 & p `1 <= (N-max Z) `1 ) )
assume Z is compact ; :: thesis: ( (N-min Z) `1 <= p `1 & p `1 <= (N-max Z) `1 )
then reconsider Z = Z as non empty compact Subset of (TOP-REAL 2) ;
( (N-min Z) `1 = lower_bound (proj1 | (N-most Z)) & (N-max Z) `1 = upper_bound (proj1 | (N-most Z)) ) by EUCLID:52;
hence ( (N-min Z) `1 <= p `1 & p `1 <= (N-max Z) `1 ) by A3, Lm3; :: thesis: verum