reserve e for set;

theorem
  for X1,X2 being set, a1,a2 being set holds [:X1 -->a1,X2-->a2:] = [:X1
  ,X2:] --> [a1,a2]
proof
  let X1,X2 be set, a1,a2 be set;
A2: dom([:X1,X2:] --> [a1,a2]) = [:dom(X1 -->a1), dom(X2-->a2):];
  now
    let x,y be object;
    assume
A3: x in dom(X1-->a1) & y in dom(X2-->a2);
    then [x,y] in dom([:X1,X2:] --> [a1,a2]) by ZFMISC_1:87;
    then
A4: [x,y] in [:X1,X2:];
    (X1-->a1).x = a1 & (X2-->a2).y = a2 by A3,FUNCOP_1:7;
    hence ([:X1,X2:] --> [a1,a2]).(x,y) = [(X1-->a1).x,(X2-->a2).y] by A4,
FUNCOP_1:7;
  end;
  hence thesis by A2,FUNCT_3:def 8;
end;
