reserve phi,fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  f,g for Function,
  X for set,
  x,y,z for object;
reserve f1,f2 for Ordinal-Sequence;

theorem
  1 in C & A <> {} & A is limit_ordinal implies for fi st dom fi = A &
  for B st B in A holds fi.B = exp(C,B) holds exp(C,A) = sup fi
proof
  assume that
A1: 1 in C and
A2: A <> {} and
A3: A is limit_ordinal;
  let fi;
  assume that
A4: dom fi = A and
A5: for B st B in A holds fi.B = exp(C,B);
  fi is increasing by A1,A4,A5,Th25;
  then lim fi = sup fi by A2,A3,A4,Th8;
  hence thesis by A2,A3,A4,A5,ORDINAL2:45;
end;
