let X be set ; :: thesis: for T being Tolerance of X
for x, y being object holds
( y in neighbourhood (,) iff [x,y] in T )
let T be Tolerance of X; :: thesis: for x, y being object holds
( y in neighbourhood (,) iff [x,y] in T )
let x, y be object ; :: thesis: ( y in neighbourhood (,) iff [x,y] in T )
hereby :: thesis: ( [x,y] in T implies y in neighbourhood (,) )
assume y in neighbourhood (,) ; :: thesis: [x,y] in T
then [y,x] in T by EQREL_1:19;
hence [x,y] in T by EQREL_1:6; :: thesis: verum
end;
assume [x,y] in T ; :: thesis: y in neighbourhood (,)
then [y,x] in T by EQREL_1:6;
hence y in neighbourhood (,) by EQREL_1:19; :: thesis: verum