let X be open Subset of REAL; :: thesis: for r being Real st r in X holds
ex g being Real st
( 0 < g & ].(r - g),(r + g).[ c= X )
let r be Real; :: thesis: ( r in X implies ex g being Real st
( 0 < g & ].(r - g),(r + g).[ c= X ) )
assume r in X ; :: thesis: ex g being Real st
( 0 < g & ].(r - g),(r + g).[ c= X )
then consider N being Neighbourhood of r such that
A1: N c= X by Th18;
consider g being Real such that
A2: ( 0 < g & N = ].(r - g),(r + g).[ ) by Def6;
take g ; :: thesis: ( 0 < g & ].(r - g),(r + g).[ c= X )
thus ( 0 < g & ].(r - g),(r + g).[ c= X ) by A1, A2; :: thesis: verum