equal
deleted
inserted
replaced
1735 obtain y where x: "x = p y" and y: "0 \<le> y" "y \<le> 1" |
1735 obtain y where x: "x = p y" and y: "0 \<le> y" "y \<le> 1" |
1736 using x by (auto simp: path_image_def) |
1736 using x by (auto simp: path_image_def) |
1737 show ?thesis |
1737 show ?thesis |
1738 unfolding path_component_def |
1738 unfolding path_component_def |
1739 proof (intro exI conjI) |
1739 proof (intro exI conjI) |
1740 have "continuous_on {0..1} (p \<circ> (( *) y))" |
1740 have "continuous_on {0..1} (p \<circ> ((*) y))" |
1741 apply (rule continuous_intros)+ |
1741 apply (rule continuous_intros)+ |
1742 using p [unfolded path_def] y |
1742 using p [unfolded path_def] y |
1743 apply (auto simp: mult_le_one intro: continuous_on_subset [of _ p]) |
1743 apply (auto simp: mult_le_one intro: continuous_on_subset [of _ p]) |
1744 done |
1744 done |
1745 then show "path (\<lambda>u. p (y * u))" |
1745 then show "path (\<lambda>u. p (y * u))" |