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

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

let i be object ; :: thesis: ( i in I implies ({x} (\/) {x,y}) . i = {x,y} . i )
assume A1: i in I ; :: thesis: ({x} (\/) {x,y}) . i = {x,y} . i
hence ({x} (\/) {x,y}) . i = ({x} . i) \/ ({x,y} . i) by PBOOLE:def 4
.= {(x . i)} \/ ({x,y} . i) by A1, Def1
.= {(x . i)} \/ {(x . i),(y . i)} by A1, Def2
.= {(x . i),(y . i)} by ZFMISC_1:9
.= {x,y} . i by A1, Def2 ;
:: thesis: verum

end;

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