reserve a,b,p,k,l,m,n,s,h,i,j,t,i1,i2 for natural Number;

theorem Th45:
  i-'2=i-'1-'1
proof
  per cases;
  suppose
A1: i>=2;
    then 1<=i by XXREAL_0:2;
    then i-1>=0 by XREAL_1:48;
    then
A2: i-'1=i-1 by XREAL_0:def 2;
    i-1>=1+1-1 by A1,XREAL_1:9;
    then i-1-1>=1-1 by XREAL_1:9;
    then i-'1-'1 =i-2 by A2,XREAL_0:def 2;
    hence thesis by XREAL_0:def 2;
  end;
  suppose
A3: i<2;
    then i-2<2-2 by XREAL_1:9;
    then
A4: i-'2=0 by XREAL_0:def 2;
A5: i+1-1<=1+1-1 by A3,NAT_1:13;
    now per cases;
      case 1<=i;
        then i=1 by A5,XXREAL_0:1;
        then i-'1-'1=0-'1 by XREAL_1:232;
        hence thesis by A4,Th35;
      end;
      case i<1;
        then i-1<1-1 by XREAL_1:9;
        then i-'1-'1=0-'1 by XREAL_0:def 2;
        hence thesis by A4,Th35;
      end;
    end;
    hence thesis;
  end;
end;
