reserve k,n,m,l,p for Nat;
reserve n0,m0 for non zero Nat;
reserve f for FinSequence;
reserve x,X,Y for set;
reserve f1,f2,f3 for FinSequence of REAL;
reserve n1,n2,m1,m2 for Nat;
reserve I,j for set;
reserve f,g for Function of I, NAT;
reserve J,K for finite Subset of I;

theorem Th32:
  m divides n0 & k divides n0 & n0<>m & n0<>k & m<>1 & k<>1 & m<>k
  implies 1+m+k+n0 <= sigma n0
proof
  assume that
A1: m divides n0 and
A2: k divides n0 and
A3: n0<>m and
A4: n0<>k and
A5: m<>1 and
A6: k<>1 and
A7: m<>k;
  per cases;
  suppose n0 = 1;
    hence thesis by A1,A3,WSIERP_1:15;
  end;
  suppose
A8: n0 <> 1;
    reconsider X2 = {m,k,n0} as finite Subset of NAT by Lm2;
    set X1 = {1};
    now
      let x be object;
      assume
A9:   x in X1 /\ X2;
      then x in X1 by XBOOLE_0:def 4;
      then
A10:  x = 1 by TARSKI:def 1;
      x in X2 by A9,XBOOLE_0:def 4;
      hence contradiction by A5,A6,A8,A10,ENUMSET1:def 1;
    end;
    then X1 /\ X2 = {} by XBOOLE_0:def 1;
    then
A11: X1 misses X2 by XBOOLE_0:def 7;
    reconsider X5 = {m} as finite Subset of NAT by Th4;
    reconsider X4 = {n0} as finite Subset of NAT by Th4;
    reconsider X6 = {k} as finite Subset of NAT by Th4;
    reconsider X3 = {m,k} as finite Subset of NAT by Th5;
    reconsider X = {1,m,k,n0} as finite Subset of NAT by Lm3;
    set Y = NatDivisors(n0) \ X;
A12: 0 + Sum((EXP 1)|X) <= Sum((EXP 1)|Y) + Sum((EXP 1)|X) by XREAL_1:7;
    now
      let x be object;
      assume
A13:  x in X5 /\ X6;
      then x in X5 by XBOOLE_0:def 4;
      then
A14:  x = m by TARSKI:def 1;
      x in X6 by A13,XBOOLE_0:def 4;
      hence contradiction by A7,A14,TARSKI:def 1;
    end;
    then X5 /\ X6 = {} by XBOOLE_0:def 1;
    then
A15: X3 = X5 \/ X6 & X5 misses X6 by ENUMSET1:1,XBOOLE_0:def 7;
    for x being object st x in X holds x in NatDivisors n0
    proof
      let x be object;
      assume
A16:  x in X;
      then reconsider x9=x as Element of NAT;
      x = 1 or x = m or x = k or x = n0 by A16,ENUMSET1:def 2;
      then x9 <> 0 & x9 divides n0 by A1,A2,INT_2:3,NAT_D:6;
      hence x in NatDivisors n0;
    end;
    then X c= NatDivisors n0;
    then NatDivisors n0 = X \/ Y by XBOOLE_1:45;
    then
A17: sigma n0 = Sum((EXP 1)|(X \/ Y)) by Def2
      .= Sum((EXP 1)|X) + Sum((EXP 1)|Y) by Th26,XBOOLE_1:79;
    now
      let x be object;
      assume
A18:  x in X3 /\ X4;
      then x in X4 by XBOOLE_0:def 4;
      then
A19:  x = n0 by TARSKI:def 1;
      x in X3 by A18,XBOOLE_0:def 4;
      hence contradiction by A3,A4,A19,TARSKI:def 2;
    end;
    then X3 /\ X4 = {} by XBOOLE_0:def 1;
    then
A20: X2 = X3 \/ X4 & X3 misses X4 by ENUMSET1:3,XBOOLE_0:def 7;
    X = X1 \/ X2 by ENUMSET1:4;
    then Sum((EXP 1)|X) = Sum((EXP 1)|X1) + Sum((EXP 1)|X2) by A11,Th26
      .= (EXP 1).1 + Sum((EXP 1)|X2) by Th27
      .= (EXP 1).1 + (Sum((EXP 1)|X3) + Sum((EXP 1)|X4)) by A20,Th26
      .= (EXP 1).1 + (Sum((EXP 1)|X5) + Sum((EXP 1)|X6) + Sum((EXP 1)|X4))
    by A15,Th26
      .= (EXP 1).1 + ((EXP 1).m + Sum((EXP 1)|X6) + Sum((EXP 1)|X4)) by Th27
      .= ((EXP 1).1 + (EXP 1).m) + (Sum((EXP 1)|X6) + Sum((EXP 1)|X4))
      .= ((EXP 1).1 + (EXP 1).m) + ((EXP 1).k + Sum((EXP 1)|X4)) by Th27
      .= ((EXP 1).1 + (EXP 1).m + (EXP 1).k) + Sum((EXP 1)|X4)
      .= (EXP 1).1 + (EXP 1).m + (EXP 1).k + (EXP 1).n0 by Th27
      .= 1|^1 + (EXP 1).m + (EXP 1).k + (EXP 1).n0 by Def1
      .= 1|^1 + m|^1 + (EXP 1).k + (EXP 1).n0 by Def1
      .= 1|^1 + m|^1 + k|^1 + (EXP 1).n0 by Def1
      .= 1|^1 + m|^1 + k|^1 + n0|^1 by Def1
      .= 1 + m|^1 + k|^1 + n0|^1
      .= 1 + m + k|^1 + n0|^1
      .= 1 + m + k + n0|^1
      .= 1 + m + k + n0;
    hence 1+m+k+n0 <= sigma n0 by A17,A12;
  end;
end;
