
theorem Th48:
  for A being Ordinal-Sequence, b, c being Ordinal
  holds b -exponent (A | c) = (b -exponent A) | c
proof
  let A be Ordinal-Sequence, b, c be Ordinal;
  A1: dom(b -exponent (A | c)) = dom(A|c) by Def1
    .= dom A /\ c by RELAT_1:61
    .= dom(b -exponent A) /\ c by Def1
    .= dom((b -exponent A) | c) by RELAT_1:61;
  now
    let x be object;
    assume A2: x in dom(b -exponent (A | c));
    then A3: x in dom(A|c) by Def1;
    then A4: x in dom A by RELAT_1:57;
    thus (b -exponent (A | c)).x = b -exponent ((A|c).x) by A3, Def1
      .= b -exponent (A.x) by A3, FUNCT_1:47
      .= (b -exponent A).x by A4, Def1
      .= ((b -exponent A)|c).x by A1, A2, FUNCT_1:47;
  end;
  hence thesis by A1, FUNCT_1:2;
end;
