How do I prove or disprove above claim ? These are the definitions of Completely normal space, arcwise connected space and mutually separated sets.
Normal and Completely Normal spaces
Mutually Separated: Two sets A and B of M are said to be mutually separated if they are disjoint and open in their union.
The definition of being completely normal is that two completely separated subsets of $X$ can be separated by disjoint neighbourhoods, where your stated definition that $A$ and $B$ are mutually separated iff they are open and disjoint in their union is equivalent to the more standard one that $A$ and $B$ are totally separated, I.e. $A \cap \overline{B} = \emptyset = \overline{A} \cap B$ (that Wikipedia also uses). A small moment's thought will convince you of this.
So the arcwise connected bit is a red herring (i.e. totally unrelated), and I suppose this must be a detail in a larger proof that you were stuck on (?), but I'm afraid it's just a restatement of a definition..