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);

theorem Th21:
  (union cod l <> {} or union dom l = {}) & l`2 is Function of
union dom l,union cod l &
for x,y st {x,y} in dom l holds {(l`2).x,(l`2).y} in cod
  l
proof
  consider g,C1,C2 such that
A1: l = [[C1,C2],g] and
A2: ( union C2={} implies union C1 = {})& g is Function of union C1,
  union C2 and
A3: for x,y st {x,y} in C1 holds {g.x,g.y} in C2 by Th18;
A4: C2 = cod l by A1;
A5: g = l`2 & C1 = dom l by A1;
  thus (union cod l <> {} or union dom l = {}) & l`2 is Function of union dom
  l,union cod l by A1,A2;
  let x,y;
  assume {x,y} in dom l;
  hence thesis by A3,A5,A4;
end;
