theorem Th70:
  dom f c= A \/ B implies f|A +* f|B = f
proof
A1: dom(f|A) = dom f /\ A & dom(f|B) = dom f /\ B by RELAT_1:61;
A2: for x being object holds
x in dom(f|A) \/ dom(f|B) implies (x in dom(f|B) implies f.x = f|B.x) &
  (not x in dom(f|B) implies f.x = f|A.x)
  proof let x be object;
    assume
A3: x in dom(f|A) \/ dom(f|B);
    thus x in dom(f|B) implies f.x = f|B.x by FUNCT_1:47;
    assume not x in dom(f|B);
    then x in dom(f|A) by A3,XBOOLE_0:def 3;
    hence thesis by FUNCT_1:47;
  end;
  assume dom f c= A \/ B;
  then dom f = dom f /\ (A \/ B) by XBOOLE_1:28
    .= dom(f|A) \/ dom(f|B) by A1,XBOOLE_1:23;
  hence thesis by A2,Def1;
end;
