I would like to ask 2 questions.
1)Can we omit 'that' when using this style?
I will fix your car on condition you wash mine.
2) When you are talking about the future, we do not use will after unless/as long as/ provided/ providing/ so long as/ whereas. (In the main clause)
Can we include 'on condition that and suppose/supposing that' also?
On condition that I will..........
Suppose that I will............
To me it doesn't sound right.
I think it'd sound better if you said, On the condition you wash mine.
More less substituting a word for a word.
I think saying "on condition you wash mine" is too much of an open statement.
Yeah, it was a little vague...By open statement, I meant further interpretation. "On Condition" could be interpreted in a few different ways, but "On the condition" implies a pre determined condition accepted. I feel it fits better than on "That condition" as it also feels a bit like an open statement...I'm probably just rambling though....
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I also find "on condition" to be strange. I would say "...on the condition that you..." Perhaps it's different over here.