let G be Group; :: thesis: for N, H being Subgroup of G

for x being Element of G st x in N ` H holds

x * N c= carr H

let N, H be Subgroup of G; :: thesis: for x being Element of G st x in N ` H holds

x * N c= carr H

let x be Element of G; :: thesis: ( x in N ` H implies x * N c= carr H )

assume x in N ` H ; :: thesis: x * N c= carr H

then ex x1 being Element of G st

( x1 = x & x1 * N c= carr H ) ;

hence x * N c= carr H ; :: thesis: verum

for x being Element of G st x in N ` H holds

x * N c= carr H

let N, H be Subgroup of G; :: thesis: for x being Element of G st x in N ` H holds

x * N c= carr H

let x be Element of G; :: thesis: ( x in N ` H implies x * N c= carr H )

assume x in N ` H ; :: thesis: x * N c= carr H

then ex x1 being Element of G st

( x1 = x & x1 * N c= carr H ) ;

hence x * N c= carr H ; :: thesis: verum