reserve x,y,z,a,b,c,X,A for set;
reserve C,D for Coherence_Space;
reserve T for Tolerance of union C;
reserve E for Tolerance of X;
reserve C,C1,C2 for Element of CSp(X);
reserve g for Element of FuncsC(X);
reserve l,l1,l2,l3 for Element of MapsC(X);
reserve T,T1,T2 for Element of TOL(X);
reserve f for Element of FuncsT(X);
reserve m,m1,m2,m3 for Element of MapsT(X);

theorem Th46:
  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)`2,(cod m)`2 by Th42;
  then
A2: rng m`2 c= (cod m)`2 by RELAT_1:def 19;
  (cod m)`2 <> {} or (dom m)`2 = {} by Th42;
  then
A3: dom m`2 = (dom m)`2 by A1,FUNCT_2:def 1;
A4: cod i1 = dom m;
  then
A5: cod(m*i1) = cod m by Th43;
  (m*i1)`2 = m`2*i1`2 & dom(m*i1) = dom i1 by A4,Th43;
  hence m*i1 = m by A3,A5,Lm4,RELAT_1:52;
A6: dom i2 = cod m;
  then
A7: cod(i2*m) = cod i2 by Th43;
  (i2*m)`2 = i2`2*m`2 & dom(i2*m) = dom m by A6,Th43;
  hence thesis by A2,A7,Lm4,RELAT_1:53;
end;
