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

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

let i be object ; :: 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_1:26
.= {x,y} . i by A1, Def2 ;
:: thesis: verum

end;

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