let x1, x2, x3, x4, x5 be set ; :: thesis: {x1,x2,x3,x4,x5} = {x1,x2,x3} \/ {x4,x5}

now

let x be set ; :: thesis: ( x in {x1,x2,x3,x4,x5} iff x in {x1,x2,x3} \/ {x4,x5} )
( x in {x1,x2,x3,x4,x5} iff ( x = x1 or x = x2 or x = x3 or x = x4 or x = x5 ) ) by Def3;
then ( x in {x1,x2,x3,x4,x5} iff ( x in {x1,x2,x3} or x in {x4,x5} ) ) by Def1, TARSKI:def 2;
hence ( x in {x1,x2,x3,x4,x5} iff x in {x1,x2,x3} \/ {x4,x5} ) by XBOOLE_0:def 3; :: thesis: verum

end;

hence {x1,x2,x3,x4,x5} = {x1,x2,x3} \/ {x4,x5} by TARSKI:2; :: thesis: verum