let M be non empty MetrStruct ; :: thesis: for a being Real holds dist_toler (M,a) = low_toler ( the distance of M,a)
let a be Real; :: thesis: dist_toler (M,a) = low_toler ( the distance of M,a)
now :: thesis: for z being object st z in low_toler ( the distance of M,a) holds
z in dist_toler (M,a)
let z be object ; :: thesis: ( z in low_toler ( the distance of M,a) implies z in dist_toler (M,a) )
assume A1: z in low_toler ( the distance of M,a) ; :: thesis: z in dist_toler (M,a)
consider x, y being object such that
A2: ( x in the carrier of M & y in the carrier of M ) and
A3: z = [x,y] by ;
reconsider x1 = x, y1 = y as Element of M by A2;
dist (x1,y1) = the distance of M . (x1,y1) by METRIC_1:def 1;
then dist (x1,y1) <= a by A1, A3, Def3;
then x1,y1 are_in_tolerance_wrt a ;
hence z in dist_toler (M,a) by ; :: thesis: verum
end;
then A4: low_toler ( the distance of M,a) c= dist_toler (M,a) ;
now :: thesis: for z being object st z in dist_toler (M,a) holds
z in low_toler ( the distance of M,a)
let z be object ; :: thesis: ( z in dist_toler (M,a) implies z in low_toler ( the distance of M,a) )
assume A5: z in dist_toler (M,a) ; :: thesis: z in low_toler ( the distance of M,a)
consider x, y being object such that
A6: ( x in the carrier of M & y in the carrier of M ) and
A7: z = [x,y] by ;
reconsider x1 = x, y1 = y as Element of M by A6;
( the distance of M . (x1,y1) = dist (x1,y1) & x1,y1 are_in_tolerance_wrt a ) by ;
then the distance of M . (x1,y1) <= a ;
hence z in low_toler ( the distance of M,a) by ; :: thesis: verum
end;
then dist_toler (M,a) c= low_toler ( the distance of M,a) ;
hence dist_toler (M,a) = low_toler ( the distance of M,a) by ; :: thesis: verum