reserve a,x,y for object, A,B for set,
  l,m,n for Nat;
reserve X,Y for set, x for object,
  p,q for Function-yielding FinSequence,
  f,g,h for Function;
reserve m,n,k for Nat, R for Relation;
reserve i,j for Nat;
reserve F for Function,
  e,x,y,z for object;
reserve a,b,c for set;
reserve A,B,I for set, X,Y for ManySortedSet of I;

theorem
 f c= g & not x in dom f implies f c= g+*(x,y)
 proof assume that
A1: f c= g;
  assume not x in dom f;
  then
A2: f c= f +*(x .--> y) by FUNCT_4:107;
  per cases;
  suppose x in dom g;
   then g+*(x,y) = g+*(x.-->y) by Def2;
   then f +*(x .--> y) c= g+*(x,y) by A1,FUNCT_4:123;
  hence f c= g+*(x,y) by A2;
  end;
  suppose not x in dom g;
  hence f c= g+*(x,y) by A1,Def2;
  end;
 end;
