reserve p,q,x,x1,x2,y,y1,y2,z,z1,z2 for set;
reserve A,B,V,X,X1,X2,Y,Y1,Y2,Z for set;
reserve C,C1,C2,D,D1,D2 for non empty set;

theorem
  for f,g being Function holds <:f,g:>.:A c= [:f.:A,g.:A:]
proof
  let f,g be Function;
  let y be object;
  assume y in <:f,g:>.:A;
  then consider x being object such that
A1: x in dom <:f,g:> and
A2: x in A and
A3: y = <:f,g:>.x by FUNCT_1:def 6;
A4: x in dom f /\ dom g by A1,Def7;
  then x in dom f by XBOOLE_0:def 4;
  then
A5: f.x in f.:A by A2,FUNCT_1:def 6;
  x in dom g by A4,XBOOLE_0:def 4;
  then
A6: g.x in g.: A by A2,FUNCT_1:def 6;
  y = [f.x,g.x] by A1,A3,Def7;
  hence thesis by A5,A6,ZFMISC_1:def 2;
end;
