reserve V for non empty set,
  A,B,A9,B9 for Element of V;
reserve f,f9 for Element of Funcs(V);
reserve m,m1,m2,m3,m9 for Element of Maps V;

theorem Th14:
  m*(id$ dom m) = m & (id$ cod m)*m = m
proof
  set i1 = id$ dom m, i2 = id$ cod m;
A1: m`2 is Function of dom m,cod m by Th9;
  then
A2: rng m`2 c= cod m by RELAT_1:def 19;
  cod m <> {} or dom m = {} by Th9;
  then
A3: dom m`2 = dom m by A1,FUNCT_2:def 1;
A4: cod i1 = dom m;
  then
A5: cod(m*i1) = cod m by Th12;
  (m*i1)`2 = m`2*i1`2 & dom(m*i1) = dom i1 by A4,Th12;
  hence m*i1 = m by A3,A5,Lm2,RELAT_1:52;
A6: dom i2 = cod m;
  then
A7: cod(i2*m) = cod i2 by Th12;
  (i2*m)`2 = i2`2*m`2 & dom(i2*m) = dom m by A6,Th12;
  hence thesis by A2,A7,Lm2,RELAT_1:53;
end;
