reserve A,B for Ordinal,
  K,M,N for Cardinal,
  x,x1,x2,y,y1,y2,z,u for object,X,Y,Z,X1,X2, Y1,Y2 for set,
  f,g for Function;
reserve m,n for Nat;
reserve x1,x2,x3,x4,x5,x6,x7,x8 for object;
reserve A,B,C for Ordinal,
  K,L,M,N for Cardinal,
  x,y,y1,y2,z,u for object,X,Y,Z,Z1,Z2 for set,
  n for Nat,
  f,f1,g,h for Function,
  Q,R for Relation;
reserve n,k for Nat;

theorem Th73:
  omega c= A implies 1+^A = A
proof
  deffunc f(Ordinal) = 1+^$1;
  consider phi being Ordinal-Sequence such that
A1: dom phi = omega & for B st B in omega holds phi.B = f(B)
  from ORDINAL2:sch 3;
A2: 1+^omega = sup phi by A1,Lm9,ORDINAL2:29
    .= sup rng phi by ORDINAL2:26;
A3: 1+^omega c= omega
  proof
    let B;
    assume B in 1+^omega;
    then consider C being Ordinal such that
A4: C in rng phi and
A5: B c= C by A2,ORDINAL2:21;
    consider x being object such that
A6: x in dom phi and
A7: C = phi.x by A4,FUNCT_1:def 3;
    reconsider x as Ordinal by A6;
    reconsider x9 = x as Cardinal by A1,A6;
    reconsider x as finite Ordinal by A1,A6;
A8: C = 1+^x by A1,A6,A7;
    succ x in omega by A1,A6,Lm9,ORDINAL1:28;
    then C in omega by A8,ORDINAL2:31;
    hence thesis by A5,ORDINAL1:12;
  end;
  assume omega c= A;
  then
A9: ex B st A = omega+^B by ORDINAL3:27;
  omega c= 1+^omega by ORDINAL3:24;
  then omega = 1+^omega by A3;
  hence thesis by A9,ORDINAL3:30;
end;
