reserve k,m,n for Element of NAT,
  a,X,Y for set,
  D,D1,D2 for non empty set;
reserve p,q for FinSequence of NAT;
reserve x,y,z,t for Variable;
reserve F,F1,G,G1,H,H1 for ZF-formula;

theorem Th12:
  H is atomic or ex H1 st len H1 + 1 <= len H
proof
A1: now
    let H;
    assume H is universal;
    then consider x,H1 such that
A2: H = All(x,H1);
    take H1;
A3: len <*4,x*> = 2 & <*4*>^<*x*> =<*4,x*> by FINSEQ_1:44,def 9;
A4: 1 + 1 + len H1 = 1 + (1 + len H1);
    len H = len(<*4*>^<*x*>) + len H1 by A2,FINSEQ_1:22;
    hence len H1 + 1 <= len H by A3,A4,NAT_1:11;
  end;
A5: now
    let H;
    assume H is negative;
    then consider H1 such that
A6: H = 'not' H1;
    take H1;
    len H = len <*2*> + len H1 by A6,FINSEQ_1:22;
    hence len H1 + 1 <= len H by FINSEQ_1:40;
  end;
A7: now
    let H;
    assume H is conjunctive;
    then consider H1,F1 such that
A8: H = H1 '&' F1;
    take H1;
A9: len(<*3*>^H1) = len <*3*> + len H1 & len <*3*> = 1 by FINSEQ_1:22,40;
    len H = len(<*3*>^H1) + len F1 by A8,FINSEQ_1:22;
    hence len H1 + 1 <= len H by A9,NAT_1:11;
  end;
  assume not H is atomic;
  then H is negative or H is conjunctive or H is universal by Th10;
  hence thesis by A5,A7,A1;
end;
