reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;
reserve A,B for set;
reserve x,y,i,j,k for object;
reserve x for set;
reserve x for object;

theorem
 for f1,f2,g1,g2 being Function
  st f1 c= g1 & f2 c= g2 & dom f1 misses dom g2
 holds f1 +* f2 c= g1 +* g2
 proof
  let f1,f2,g1,g2 be Function such that
A1: f1 c= g1 and
A2: f2 c= g2 and
A3: dom f1 misses dom g2;
A4: f1 c= g1 +* g2 by A1,A3,Th117;
   g2 c= g1 +* g2 by Th25;
   then f2 c= g1 +* g2 by A2;
  hence f1 +* f2 c= g1 +* g2 by A4,Th87;
 end;
