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 Th18:
  ex g,C1,C2 st l = [[C1,C2],g] & (union C2 = {} implies union C1
= {}) & g is Function of union C1,union C2 & for x,y st {x,y} in C1 holds {g.x,
  g.y} in C2
proof
  l in {[[C1,C2],g]: (union C2={} implies union C1={}) & g is Function of
  union C1,union C2 & for x,y st {x,y} in C1 holds {g.x,g.y} in C2};
  then
  ex C1,C2,g st l = [[C1,C2],g] & (union C2={} implies union C1={}) & g is
  Function of union C1,union C2 & for x,y st {x,y} in C1 holds {g.x,g.y} in C2;
  hence thesis;
end;
