reserve
  a,b,c,d,e for Ordinal,
  m,n for Nat,
  f for Ordinal-Sequence,
  x for object;
reserve S,S1,S2 for Sequence;

theorem Th2:
  dom S = a+^b implies ex S1,S2 st S = S1^S2 & dom S1 = a & dom S2 = b
  proof assume
A1: dom S = a+^b;
    set S1 = S|a;
A2: a c= a+^b by ORDINAL3:24; then
A3: dom S1 = a by A1,RELAT_1:62;
    deffunc F(Ordinal) = S.(a+^$1);
    consider S2 such that
A4: dom S2 = b & for c st c in b holds S2.c = F(c) from ORDINAL2:sch 2;
    take S1,S2; set s = S1^S2;
A5: dom S = dom s by A1,A3,A4,ORDINAL4:def 1;
    now let x be object; assume
A6:   x in dom S; then reconsider z = x as Ordinal;
      per cases by A1,A6,ORDINAL3:38;
      suppose
A7:     z in a;
        hence S.x = S1.z by FUNCT_1:49 .= s.x by A7,A3,ORDINAL4:def 1;
      end;
      suppose ex c st c in b & z = a+^c; then
        consider c such that
A8:     c in b & z = a+^c;
        thus S.x = S2.c by A4,A8 .= s.x by A8,A3,A4,ORDINAL4:def 1;
      end;
    end;
    hence thesis by A2,A4,A5,A1,FUNCT_1:2,RELAT_1:62;
  end;
