let I be set ; :: thesis: for X being ManySortedSet of I st ( for i being set st i in I holds
X . i = {} ) holds
X = [[0]] I
let X be ManySortedSet of I; :: thesis: ( ( for i being set st i in I holds
X . i = {} ) implies X = [[0]] I )
assume A1: for i being set st i in I holds
X . i = {} ; :: thesis: X = [[0]] I

now

let i be set ; :: thesis: ( i in I implies X . i = ([[0]] I) . i )
assume i in I ; :: thesis: X . i = ([[0]] I) . i
hence X . i = {} by A1
.= ([[0]] I) . i by Th5 ;
:: thesis: verum

end;

hence X = [[0]] I by Th3; :: thesis: verum