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

now

let i be set ; :: 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 7
.= {(x . i)} \/ ({x,y} . i) by A1, Def1
.= {(x . i)} \/ {(x . i),(y . i)} by A1, Def2
.= {(x . i),(y . i)} by ZFMISC_1:14
.= {x,y} . i by A1, Def2 ;
:: thesis: verum

end;

hence {x} \/ {x,y} = {x,y} by PBOOLE:3; :: thesis: verum