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 Th37:
  n > 4 implies ex k being Nat st n = 2*k & k > 2 or n = 2*k+1 & k > 1
  proof
    assume
A1: n > 4;
    per cases;
    suppose n is even;
      then consider k being Nat such that
A2:   n = 2*k;
      take k;
      4 = 2*2;
      hence thesis by A1,A2,XREAL_1:66;
    end;
    suppose n is odd;
      then consider k such that
A3:   n = 2*k+1 by ABIAN:9;
      take k;
      now
        assume k <= 1;
        then 2*k <= 2*1 by XREAL_1:64;
        then 2*k+1 <= 2+1 by XREAL_1:6;
        hence contradiction by A1,A3,XXREAL_0:2;
      end;
      hence thesis by A3;
    end;
  end;
