reserve a,b,c,d for Ordinal;
reserve l for non empty limit_ordinal Ordinal;
reserve u for Element of l;
reserve A for non empty Ordinal;
reserve e for Element of A;
reserve X,Y,x,y,z for set;
reserve n,m for Nat;
reserve f for Ordinal-Sequence;
reserve U,W for Universe;
reserve F,phi for normal Ordinal-Sequence of W;
reserve g for Ordinal-Sequence-valued Sequence;

theorem Th71:
  a c= b iff c-Veblen a c= c-Veblen b
  proof
    set U = Tarski-Class(c\/b\/omega);
    set W = Tarski-Class(c\/a\/omega);
A1: n in omega & omega in U & omega in W by Th57,ORDINAL1:def 12;
A2: b in U & c in U by Th66;
A3: a in W & c in W by Th66;
    reconsider f = U-Veblen.c as increasing Ordinal-Sequence of U
    by A1,Th66,Th62;
    reconsider g = W-Veblen.c as increasing Ordinal-Sequence of W
    by A1,Th66,Th62;
A4: dom f = On U & dom g = On W by FUNCT_2:def 1;
A5: b in On U & a in On W by A2,A3,ORDINAL1:def 9;
    hereby
      assume
A6:   a c= b; then
A7:   a in U by A2,CLASSES1:def 1;
      per cases by A6;
      suppose a = b;
        hence c-Veblen a c= c-Veblen b;
      end;
      suppose a c< b; then
        a in b by ORDINAL1:11; then
        f.a in f.b by A4,A5,ORDINAL2:def 12; then
        c-Veblen a in c-Veblen b by A7,A1,A2,A3,Th64;
        hence c-Veblen a c= c-Veblen b by ORDINAL1:def 2;
      end;
    end;
    assume
A8: c-Veblen a c= c-Veblen b & a c/= b; then
A9: b in a by ORDINAL1:16; then
A10: b in W by A3,ORDINAL1:10;
    g.b in g.a by A4,A5,A9,ORDINAL2:def 12; then
    c-Veblen b in c-Veblen a by A1,A2,A3,A10,Th64; then
    c-Veblen b in c-Veblen b by A8;
    hence thesis;
  end;
