reserve T,U for non empty TopSpace;
reserve t for Point of T;
reserve n for Nat;
reserve T for TopStruct;
reserve f for PartFunc of R^1, T;
reserve c for Curve of T;
reserve T for non empty TopStruct;

theorem Th37:
  for c being with_endpoints Curve of T
  for r being Real holds
  ex c1,c2 being Element of Curves T
  st c = c1 + c2 & c1 = c | [.inf dom c, r.] & c2 = c | [.r, sup dom c.]
  proof
    let c be with_endpoints Curve of T;
    let r be Real;
    set c1 = c | [.inf dom c, r.];
    set c2 = c | [.r, sup dom c.];
    reconsider c1 as Curve of T by Th26;
    reconsider c2 as Curve of T by Th26;
    take c1,c2;
    c1 \/ c2 = c
    proof
      per cases;
      suppose
A1:     r < inf dom c;
        then [.inf dom c, r.] = {} by XXREAL_1:29;
        then
A2:     c1 = {};
        [.inf dom c, sup dom c.] c= [.r, sup dom c.] by A1,XXREAL_1:34;
        then dom c c= [.r, sup dom c.] by Th27;
        hence thesis by A2,RELAT_1:68;
      end;
      suppose
A3:     r >= inf dom c;
        per cases;
        suppose
A4:      r > sup dom c;
          then [.r, sup dom c.] = {} by XXREAL_1:29;
          then
A5:       c2 = {};
          [.inf dom c, sup dom c.] c= [.inf dom c, r.] by A4,XXREAL_1:34;
          then dom c c= [.inf dom c, r.] by Th27;
          hence thesis by A5,RELAT_1:68;
        end;
        suppose
A6:       r <= sup dom c;
          dom c = [.inf dom c, sup dom c.] by Th27
          .= [.inf dom c,r.] \/ [.r, sup dom c.] by A6,A3,XXREAL_1:165;
          then c | dom c
          = c1 \/ c2 by RELAT_1:78;
          hence thesis;
        end;
      end;
    end;
    hence thesis by Def12;
  end;
