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 Th42:
  ((cod m)`2 <> {} or (dom m)`2 = {}) & m`2 is Function of (dom m)
  `2,(cod m)`2 & for x,y st [x,y] in (dom m)`1 holds [m`2.x,m`2.y] in (cod m)`1
proof
  consider f,T1,T2 such that
A1: m = [[T1,T2],f] and
A2: ( T2`2 = {} implies T1`2 = {})& f is Function of T1`2,T2`2 and
A3: for x,y st [x,y] in T1`1 holds [f.x,f.y] in T2`1 by Th39;
A4: T2 = cod m by A1;
A5: f = m`2 & T1 = dom m by A1;
  thus
  ((cod m)`2<>{} or (dom m)`2={} ) & m`2 is Function of (dom m)`2,(cod m)
  `2 by A1,A2;
  let x,y;
  assume [x,y] in (dom m)`1;
  hence thesis by A3,A5,A4;
end;
