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 Th19:
  for f being Function of union C1,union C2 st (union C2={}
implies union C1={}) & (for x,y st {x,y} in C1 holds {f.x,f.y} in C2) holds [[
  C1,C2],f] in MapsC(X)
proof
  let f be Function of union C1,union C2;
  assume that
A1: union C2={} implies union C1={} and
A2: for x,y st {x,y} in C1 holds {f.x,f.y} in C2;
  reconsider f9 = f as Element of FuncsC(X) by A1,Th17;
  for x,y st {x,y} in C1 holds {f9.x,f9.y} in C2 by A2;
  hence thesis by A1;
end;
