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 Th31:
  m divides n0 & n0<>m & m<>1 implies 1+m+n0 <= sigma n0
proof
  assume
A1: m divides n0;
  assume
A2: n0<>m;
  assume
A3: m<>1;
  per cases;
  suppose
    n0 = 1;
    hence thesis by A1,A2,WSIERP_1:15;
  end;
  suppose
A4: n0 <> 1;
    reconsider X2 = {m,n0} as finite Subset of NAT by Th5;
    set X1 = {1};
    now
      let x be object;
      assume
A5:   x in X1 /\ X2;
      then x in X1 by XBOOLE_0:def 4;
      then
A6:   x = 1 by TARSKI:def 1;
      x in X2 by A5,XBOOLE_0:def 4;
      hence contradiction by A3,A4,A6,TARSKI:def 2;
    end;
    then X1 /\ X2 = {} by XBOOLE_0:def 1;
    then
A7: X1 misses X2 by XBOOLE_0:def 7;
    reconsider X4 = {n0} as finite Subset of NAT by Th4;
    reconsider X3 = {m} as finite Subset of NAT by Th4;
    reconsider X = {1,m,n0} as finite Subset of NAT by Lm2;
    set Y = NatDivisors(n0) \ X;
A8: 0 + Sum((EXP 1)|X) <= Sum((EXP 1)|Y) + Sum((EXP 1)|X) by XREAL_1:7;
    for x being object st x in X holds x in NatDivisors n0
    proof
      let x be object;
      assume
A9:   x in X;
      then reconsider x9=x as Element of NAT;
      x = 1 or x = m or x = n0 by A9,ENUMSET1:def 1;
      then x9 <> 0 & x9 divides n0 by A1,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
A10: 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
A11:  x in X3 /\ X4;
      then x in X3 by XBOOLE_0:def 4;
      then
A12:  x = m by TARSKI:def 1;
      x in X4 by A11,XBOOLE_0:def 4;
      hence contradiction by A2,A12,TARSKI:def 1;
    end;
    then X3 /\ X4 = {} by XBOOLE_0:def 1;
    then
A13: X2 = X3 \/ X4 & X3 misses X4 by ENUMSET1:1,XBOOLE_0:def 7;
    X = X1 \/ X2 by ENUMSET1:2;
    then Sum((EXP 1)|X) = Sum((EXP 1)|X1) + Sum((EXP 1)|X2) by A7,Th26
      .= (EXP 1).1 + Sum((EXP 1)|X2) by Th27
      .= (EXP 1).1 + (Sum((EXP 1)|X3) + Sum((EXP 1)|X4)) by A13,Th26
      .= (EXP 1).1 + ((EXP 1).m + Sum((EXP 1)|X4)) by Th27
      .= ((EXP 1).1 + (EXP 1).m) + Sum((EXP 1)|X4)
      .= (EXP 1).1 + (EXP 1).m + (EXP 1).n0 by Th27
      .= 1|^1 + (EXP 1).m + (EXP 1).n0 by Def1
      .= 1|^1 + m|^1 + (EXP 1).n0 by Def1
      .= 1|^1 + m|^1 + n0|^1 by Def1
      .= 1 + m|^1 + n0|^1
      .= 1 + m + n0|^1
      .= 1 + m + n0;
    hence 1+m+n0 <= sigma n0 by A10,A8;
  end;
end;
