let I be set ; :: thesis: for X, x, y being ManySortedSet of I st ( X = [0] I or X = {x} or X = {y} or X = {x,y} ) holds
X \ {x,y} = [0] I
let X, x, y be ManySortedSet of I; :: thesis: ( ( X = [0] I or X = {x} or X = {y} or X = {x,y} ) implies X \ {x,y} = [0] I )
assume A1: ( X = [0] I or X = {x} or X = {y} or X = {x,y} ) ; :: thesis: X \ {x,y} = [0] I

now

let i be set ; :: thesis: ( i in I implies (X \ {x,y}) . b₁ = ([0] I) . b₁ )
assume A2: i in I ; :: thesis: (X \ {x,y}) . b₁ = ([0] I) . b₁

per cases ( X = [0] I or X = {x} or X = {y} or X = {x,y} ) by A1;

suppose X = [0] I ; :: thesis: (X \ {x,y}) . b₁ = ([0] I) . b₁

then A3: X . i = {} by PBOOLE:5;
thus (X \ {x,y}) . i = (X . i) \ ({x,y} . i) by A2, PBOOLE:def 9
.= ([0] I) . i by A3, PBOOLE:5 ; :: thesis: verum

end;

suppose X = {x} ; :: thesis: (X \ {x,y}) . b₁ = ([0] I) . b₁

then A4: X . i = {(x . i)} by A2, Def1;
thus (X \ {x,y}) . i = (X . i) \ ({x,y} . i) by A2, PBOOLE:def 9
.= (X . i) \ {(x . i),(y . i)} by A2, Def2
.= {} by A4, ZFMISC_1:75
.= ([0] I) . i by PBOOLE:5 ; :: thesis: verum

end;

suppose X = {y} ; :: thesis: (X \ {x,y}) . b₁ = ([0] I) . b₁

then A5: X . i = {(y . i)} by A2, Def1;
thus (X \ {x,y}) . i = (X . i) \ ({x,y} . i) by A2, PBOOLE:def 9
.= (X . i) \ {(x . i),(y . i)} by A2, Def2
.= {} by A5, ZFMISC_1:75
.= ([0] I) . i by PBOOLE:5 ; :: thesis: verum

end;

suppose X = {x,y} ; :: thesis: (X \ {x,y}) . b₁ = ([0] I) . b₁

then A6: X . i = {(x . i),(y . i)} by A2, Def2;
thus (X \ {x,y}) . i = (X . i) \ ({x,y} . i) by A2, PBOOLE:def 9
.= (X . i) \ {(x . i),(y . i)} by A2, Def2
.= {} by A6, ZFMISC_1:75
.= ([0] I) . i by PBOOLE:5 ; :: thesis: verum

end;

end;

end;

hence X \ {x,y} = [0] I by PBOOLE:3; :: thesis: verum