reserve a,b,c,k,m,n for Nat;
reserve i,j,x,y for Integer;
reserve p,q for Prime;
reserve r,s for Real;

theorem Th44:
  for m being positive Nat holds
  card { [x,y] where x,y is positive Nat: x+y = m+1 } = m
  proof
    let m be positive Nat;
    set A = { [x,y] where x,y is positive Nat: x+y = m+1 };
A1: Seg m,A are_equipotent
    proof
      defpred P[object,object] means
      ex k being Nat st k = $1 & $2 = [k,m-k+1];
A2:   for e being object st e in Seg m ex u being object st P[e,u]
      proof
        let e be object;
        assume e in Seg m;
        then reconsider e as Nat;
        take [e,m-e+1], e;
        thus thesis;
      end;
      consider f being Function such that
A3:   dom f = Seg m and
A4:   for e being object st e in Seg m holds P[e,f.e] from CLASSES1:sch 1(A2);
      take f;
      thus f is one-to-one
      proof
        let x1,x2 be object such that
A5:     x1 in dom f & x2 in dom f and
A6:     f.x1 = f.x2;
        reconsider x1,x2 as Element of NAT by A3,A5;
        P[x1,f.x1] & P[x2,f.x2] by A3,A4,A5;
        hence thesis by A6,XTUPLE_0:1;
      end;
      thus dom f = Seg m by A3;
      thus rng f c= A
      proof
        let b be object;
        assume b in rng f;
        then consider a being object such that
A7:     a in dom f and
A8:     f.a = b by FUNCT_1:def 3;
        consider k being Nat such that
A9:     k = a and
A10:    f.a = [k,m-k+1] by A3,A4,A7;
A11:    1 <= k by A3,A7,A9,FINSEQ_1:1;
        k <= m by A3,A7,A9,FINSEQ_1:1;
        then m-k >= m-m by XREAL_1:10;
        then
A12:    m-k+1 is positive Element of NAT by INT_1:3;
        k+(m-k+1) = m+1;
        hence thesis by A8,A10,A11,A12;
      end;
      let b be object;
      assume b in A;
      then consider x,y being positive Nat such that
A13:  b = [x,y] and
A14:  x+y = m+1;
A15:  0+1 <= x by NAT_1:13;
      x+0 < x+y by XREAL_1:6;
      then x <= m by A14,NAT_1:13;
      then
A16:  x in Seg m by A15;
      then P[x,f.x] by A4;
      then f.x = [x,m-x+1]
      .= [x,y] by A14;
      hence thesis by A3,A13,A16,FUNCT_1:def 3;
    end;
A17: Seg m,m are_equipotent by FINSEQ_1:54;
    card m = m;
    hence thesis by A1,A17,WELLORD2:15,CARD_1:5;
  end;
