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;

theorem
  for f,g being Function, A being set st f|A = g|A & f,g equal_outside A
  holds f = g
proof
  let f,g be Function, A be set such that
A1: f|A = g|A & f,g equal_outside A;
  thus f = f|(dom f \/ A) by RELAT_1:68,XBOOLE_1:7
    .= f|(dom f \ A \/ A) by XBOOLE_1:39
    .= f|(dom f \ A) \/ f|A by RELAT_1:78
    .= g|(dom g \ A) \/ g|A by A1
    .= g|(dom g \ A \/ A) by RELAT_1:78
    .= g|(dom g \/ A) by XBOOLE_1:39
    .= g by RELAT_1:68,XBOOLE_1:7;
end;
