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

now

let i be set ; :: thesis: ( i in I implies (union {{x},{y}}) . i = {x,y} . i )
assume A1: i in I ; :: thesis: (union {{x},{y}}) . i = {x,y} . i
hence (union {{x},{y}}) . i = union ({{x},{y}} . i) by MBOOLEAN:def 2
.= union {({x} . i),({y} . i)} by A1, Def2
.= union {{(x . i)},({y} . i)} by A1, Def1
.= union {{(x . i)},{(y . i)}} by A1, Def1
.= {(x . i),(y . i)} by ZFMISC_1:32
.= {x,y} . i by A1, Def2 ;
:: thesis: verum

end;

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