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- Quotient and remainder are not defined for every
`x`and`y`:(

`x`div 0) and (`x`mod 0) are undefined for every`x`. - If quotient respectively remainder exist,
they are unique.
**forall**`x`,`y`,`q`_{0},`q`_{1},`r`_{0}<`y`,`r`_{1}<`y`:( `x`=(`q`_{0}*`y`)+`r`_{0}/\`x`=(`q`_{1}*`y`)+`r`_{1}) => (`q`_{0}=`q`_{1}/\`r`_{0}=`r`_{1}). - If the divisor is not null, quotient and remainder exist:
**forall**`x`,`y`!= 0: (**exists**q,`r`:`r`<`y`/\`x`= (`q`*`y`)+`r`). - We thus have the following relationship:
**forall**`x`,`y`!= 0:`x`= (`x`div`y`)*`y`+ (`x`mod`y`).

Author: Wolfgang Schreiner

Last Modification: November 16, 1999