let I be set ; :: thesis: for x, y being ManySortedSet of I holds
( {x} \ {x,y} = [[0]] I & {y} \ {x,y} = [[0]] I )
let x, y be ManySortedSet of I; :: thesis: ( {x} \ {x,y} = [[0]] I & {y} \ {x,y} = [[0]] I )

now

let i be set ; :: thesis: ( i in I implies ({x} \ {x,y}) . i = ([[0]] I) . i )
assume A1: i in I ; :: thesis: ({x} \ {x,y}) . i = ([[0]] I) . i
hence ({x} \ {x,y}) . i = ({x} . i) \ ({x,y} . i) by PBOOLE:def 6
.= {(x . i)} \ ({x,y} . i) by A1, Def1
.= {(x . i)} \ {(x . i),(y . i)} by A1, Def2
.= {} by ZFMISC_1:16
.= ([[0]] I) . i by PBOOLE:5 ;
:: thesis: verum

end;

hence {x} \ {x,y} = [[0]] I by PBOOLE:3; :: thesis: {y} \ {x,y} = [[0]] I

now

let i be set ; :: thesis: ( i in I implies ({y} \ {x,y}) . i = ([[0]] I) . i )
assume A2: i in I ; :: thesis: ({y} \ {x,y}) . i = ([[0]] I) . i
hence ({y} \ {x,y}) . i = ({y} . i) \ ({x,y} . i) by PBOOLE:def 6
.= {(y . i)} \ ({x,y} . i) by A2, Def1
.= {(y . i)} \ {(x . i),(y . i)} by A2, Def2
.= {} by ZFMISC_1:16
.= ([[0]] I) . i by PBOOLE:5 ;
:: thesis: verum

end;

hence {y} \ {x,y} = [[0]] I by PBOOLE:3; :: thesis: verum