let C be non empty compact Subset of (TOP-REAL 2); :: thesis: ( C is horizontal iff S-bound C = N-bound C )
thus ( C is horizontal implies S-bound C = N-bound C ) :: thesis: ( S-bound C = N-bound C implies C is horizontal )

proof

A1: N-min C in C by Th11;
A2: S-min C in C by Th12;
assume A3: C is horizontal ; :: thesis: S-bound C = N-bound C
thus S-bound C = (S-min C) `2 by EUCLID:52
.= (N-min C) `2 by A3, A2, A1
.= N-bound C by EUCLID:52 ; :: thesis: verum

end;

assume A4: S-bound C = N-bound C ; :: thesis: C is horizontal
let p be Point of (TOP-REAL 2); :: according to SPPOL_1:def 2 :: thesis: for b₁ being Element of the carrier of (TOP-REAL 2) holds
( not p in C or not b₁ in C or p `2 = b<sub>1</sub> `2 )
let q be Point of (TOP-REAL 2); :: thesis: ( not p in C or not q in C or p `2 = q `2 )
assume that
A5: p in C and
A6: q in C ; :: thesis: p `2 = q `2
A7: p `2 <= N-bound C by A5, PSCOMP_1:24;
S-bound C <= p `2 by A5, PSCOMP_1:24;
then A8: p `2 = S-bound C by A4, A7, XXREAL_0:1;
A9: q `2 <= N-bound C by A6, PSCOMP_1:24;
S-bound C <= q `2 by A6, PSCOMP_1:24;
hence p `2 = q `2 by A4, A9, A8, XXREAL_0:1; :: thesis: verum