equal
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inserted
replaced
1141 obtain e where e: "e > 0" "cball z e \<inter> affine hull closure S \<le> closure S" |
1141 obtain e where e: "e > 0" "cball z e \<inter> affine hull closure S \<le> closure S" |
1142 using z rel_interior_cball[of "closure S"] by auto |
1142 using z rel_interior_cball[of "closure S"] by auto |
1143 hence *: "0 < e/norm(z-x)" using e False by auto |
1143 hence *: "0 < e/norm(z-x)" using e False by auto |
1144 define y where "y = z + (e/norm(z-x)) *\<^sub>R (z-x)" |
1144 define y where "y = z + (e/norm(z-x)) *\<^sub>R (z-x)" |
1145 have yball: "y \<in> cball z e" |
1145 have yball: "y \<in> cball z e" |
1146 using mem_cball y_def dist_norm[of z y] e by auto |
1146 using y_def dist_norm[of z y] e by auto |
1147 have "x \<in> affine hull closure S" |
1147 have "x \<in> affine hull closure S" |
1148 using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast |
1148 using x rel_interior_subset_closure hull_inc[of x "closure S"] by blast |
1149 moreover have "z \<in> affine hull closure S" |
1149 moreover have "z \<in> affine hull closure S" |
1150 using z rel_interior_subset hull_subset[of "closure S"] by blast |
1150 using z rel_interior_subset hull_subset[of "closure S"] by blast |
1151 ultimately have "y \<in> affine hull closure S" |
1151 ultimately have "y \<in> affine hull closure S" |