reserve k,n,m for Nat,
  A,B,C for Ordinal,
  X for set,
  x,y,z for object;
reserve f,g,h,fx for Function,
  K,M,N for Cardinal,
  phi,psi for
  Ordinal-Sequence;
reserve a,b for Aleph;
reserve a,b for Aleph;

theorem Th32:
  Sum (b-powerfunc_of a) c= exp(a,b)
proof
  set X = { c where c is Element of a: c is Cardinal};
  set f = X --> exp(a,b);
  X c= a
  proof
    let x be object;
    assume x in X;
    then ex c being Element of a st x = c & c is Cardinal;
    hence thesis;
  end;
  then
A1: f c= a --> exp(a,b) by FUNCT_4:4;
  Sum (a --> exp(a,b)) = a*`exp( a,b) by CARD_2:65;
  then
A2: Sum f c= a*`exp(a,b) by A1,CARD_3:33;
A3: dom f = X & dom (b-powerfunc_of a) = X
  proof
    thus dom f = X;
    thus dom (b-powerfunc_of a) c= X
    proof
      let x be object;
      assume x in dom (b-powerfunc_of a);
      then x in a & x is Cardinal by Def2;
      hence thesis;
    end;
    let x be object;
    assume x in X;
    then ex c being Element of a st x = c & c is Cardinal;
    hence thesis by Def2;
  end;
  1 in b & exp(a,1) = a by Lm1,Th15,CARD_2:27;
  then a c= exp(a,b) by CARD_2:93;
  then
A4: a*`exp(a,b) = exp(a,b) by Th17;
  now
    let x be object;
    assume
A5: x in X;
    then consider c being Element of a such that
A6: x = c and
A7: c is Cardinal;
    reconsider c as Cardinal by A7;
A8: f.x = exp(a,b) by A5,FUNCOP_1:7;
    (b-powerfunc_of a).x = exp(c,b) by A6,Def2;
    hence (b-powerfunc_of a).x c= f.x by A8,CARD_2:93;
  end;
  then Sum (b-powerfunc_of a) c= Sum f by A3,CARD_3:30;
  hence thesis by A2,A4;
end;
