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Let $\\{K_{i}{\\to} F_{i-1}{\\hookrightarrow} F_{i} \\,|\\, 1{\\le} i {\\le} n,\\, F_{0}{=} \\{\\ast\\} \\; F_{1}{=} \\Sigma{K_{1}} \\; \\text{and}\\; F_{n}{\\simeq} G \\}$ be a cone-decomposition of $G$ of length $m$ and $F'_{1}=\\Sigma{K'_{1}} \\subset F_{1}$ with $K'_{1} \\subset K_{1}$ which satisfy $F_{i}F'_{1} \\subset F_{i+1}$ up to homotopy for any $i$. 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