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

now :: thesis: for i being object st i in I holds
({x} (\) {x,y}) . i = (EmptyMS I) . i

let i be object ; :: thesis: ( i in I implies ({x} (\) {x,y}) . i = (EmptyMS I) . i )
assume A1: i in I ; :: thesis: ({x} (\) {x,y}) . i = (EmptyMS 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
.= (EmptyMS I) . i by PBOOLE:5 ;
:: thesis: verum

end;

hence {x} (\) {x,y} = EmptyMS I ; :: thesis: verum